Q.Find the unit vector in the direction of vector $\vec{PQ}$, where P and Q are the points $(1, 2, 3)$ and $(4, 5, 6)$, respectively.

Showing the 12 most recent of 20 on this concept.

1. TG EAPCET 2021Set eng-2021-08-06-AN1 markMCQ
   Q.If $\vec{P} = \vec{i} - 2\vec{j} + 3\vec{k},\ 2\vec{i} + 3\vec{j} - 4\vec{k},\ 4\vec{i} + 13\vec{j} - 18\vec{k}$ are the position vectors of three collinear points A, B, C respectively, then the vector in the direction of $\overrightarrow{AB}$ of length $|P|$ units is (A) $\dfrac{2}{5\sqrt{3}}\left(\vec{i} + 5\vec{j} - 7\vec{k}\right)$ (B) $\dfrac{1}{\sqrt{83}}\left(3\vec{i} + 5\vec{j} - 7\vec{k}\right)$ (C) $\dfrac{1}{\sqrt{78}}\left(2\vec{i} + 5\vec{j} - 7\vec{k}\right)$ (D) $\dfrac{1}{5\sqrt{3}}\left(\vec{i} + 5\vec{j} - 7\vec{k}\right)$

   ›Reveal solutionSolution

   The key idea is to find the unit vector along $\overrightarrow{AB}$ and scale it by $|\vec{P}|$; the correct option is (D).

   The problem gives three collinear points A, B, C with position vectors $\vec{P}$, $2\vec{i}+3\vec{j}-4\vec{k}$, and $4\vec{i}+13\vec{j}-18\vec{k}$ respectively. Wait — careful: $\vec{P}$ itself is the position vector of A, given as $\vec{i} - 2\vec{j} + 3\vec{k}$. So A, B, C are collinear, meaning vectors $\overrightarrow{AB}$ and $\overrightarrow{AC}$ are parallel. We need the vector in the direction of $\overrightarrow{AB}$ whose length equals $|\vec{P}|$.

   1. Find $\overrightarrow{AB}$ and $\overrightarrow{AC}$.

      $\overrightarrow{AB} = \vec{B} - \vec{A} = (2\vec{i}+3\vec{j}-4\vec{k}) - (\vec{i}-2\vec{j}+3\vec{k}) = \vec{i} + 5\vec{j} - 7\vec{k}$.

      $\overrightarrow{AC} = \vec{C} - \vec{A} = (4\vec{i}+13\vec{j}-18\vec{k}) - (\vec{i}-2\vec{j}+3\vec{k}) = 3\vec{i} + 15\vec{j} - 21\vec{k}$.

      Notice $\overrightarrow{AC} = 3(\vec{i} + 5\vec{j} - 7\vec{k}) = 3\overrightarrow{AB}$, confirming collinearity. So the direction of $\overrightarrow{AB}$ is given by the vector $\vec{i} + 5\vec{j} - 7\vec{k}$.

   2. Find the unit vector along $\overrightarrow{AB}$.

      Magnitude of $\overrightarrow{AB}$: $|\overrightarrow{AB}| = \sqrt{1^2 + 5^2 + (-7)^2} = \sqrt{1 + 25 + 49} = \sqrt{75} = 5\sqrt{3}$.

      So the unit vector is $\hat{u} = \frac{\vec{i} + 5\vec{j} - 7\vec{k}}{5\sqrt{3}}$.

   3. Find $|\vec{P}|$.

      $\vec{P} = \vec{i} - 2\vec{j} + 3\vec{k}$, so $|\vec{P}| = \sqrt{1^2 + (-2)^2 + 3^2} = \sqrt{1 + 4 + 9} = \sqrt{14}$.

      Watch out

      A common mistake is to confuse $\vec{P}$ (the position vector of A) with the vector we need to scale. The required vector has length $|\vec{P}|$, not $\vec{P}$ itself.

   4. Scale the unit vector by $|\vec{P}|$.

      The required vector = $|\vec{P}| \cdot \hat{u} = \sqrt{14} \cdot \frac{\vec{i} + 5\vec{j} - 7\vec{k}}{5\sqrt{3}} = \frac{\sqrt{14}}{5\sqrt{3}} (\vec{i} + 5\vec{j} - 7\vec{k})$. …

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2. TG EAPCET 2024Set eng-2024-05-09-FN1 markMCQ
   Q.$\vec{i} - 2\vec{j} + \vec{k},\ 2\vec{i} + \vec{j} - \vec{k},\ \vec{i} - \vec{j} - 2\vec{k}$ are the position vectors of the vertices A, B, C of a triangle ABC respectively. If D and E are the mid points of BC and CA respectively, then the unit vector along $\overrightarrow{DE}$ is (A) $\dfrac{1}{7}(3\vec{i} - 2\vec{j} + 6\vec{k})$ (B) $\dfrac{1}{\sqrt{14}}(-\vec{i} - 3\vec{j} + 2\vec{k})$ (C) $\dfrac{1}{\sqrt{3}}(\vec{i} - \vec{j} - \vec{k})$ (D) $\dfrac{1}{13}(12\vec{i} + 3\vec{j} + 4\vec{k})$

   ›Reveal solutionSolution

   The key idea is that $\overrightarrow{DE}$ is half of $\overrightarrow{AB}$ (by the midsegment theorem in vector form). Computing $\overrightarrow{AB}$ from the given position vectors and halving it gives $\frac{1}{2}(\vec{i} - 3\vec{j} + 2\vec{k})$, whose unit vector is $\frac{1}{\sqrt{14}}(-\vec{i} - 3\vec{j} + 2\vec{k})$, matching option (B).

   Concept and intuition:

   In any triangle, the segment joining the midpoints of two sides is parallel to the third side and half its length. Here, D is the midpoint of BC and E is the midpoint of CA, so DE is parallel to BA (or AB) and exactly half its length. Therefore, instead of finding D and E separately and subtracting, we can directly compute $\overrightarrow{DE} = \frac{1}{2} \overrightarrow{BA}$ (or $-\frac{1}{2}\overrightarrow{AB}$). This saves work and avoids sign errors. Then we just need the unit vector along that result.

   Step-by-step solution:

   1. Write the position vectors clearly Let

   $\vec{A} = \vec{i} - 2\vec{j} + \vec{k}, \quad \vec{B} = 2\vec{i} + \vec{j} - \vec{k}, \quad \vec{C} = \vec{i} - \vec{j} - 2\vec{k}.$

   2. Find $\overrightarrow{AB}$

   $\overrightarrow{AB} = \vec{B} - \vec{A} = (2-1)\vec{i} + (1 - (-2))\vec{j} + (-1 - 1)\vec{k} = \vec{i} + 3\vec{j} - 2\vec{k}.$

   3. Apply the midpoint theorem D is midpoint of BC, E is midpoint of CA. In vector geometry,

   $\overrightarrow{DE} = \frac{1}{2}\overrightarrow{BA} = -\frac{1}{2}\overrightarrow{AB}.$

   So

   $\overrightarrow{DE} = -\frac{1}{2}(\vec{i} + 3\vec{j} - 2\vec{k}) = -\frac{1}{2}\vec{i} - \frac{3}{2}\vec{j} + \vec{k}.$

   4. Find the magnitude of $\overrightarrow{DE}$

   $|\overrightarrow{DE}| = \sqrt{\left(-\frac{1}{2}\right)^2 + \left(-\frac{3}{2}\right)^2 + (1)^2} = \sqrt{\frac{1}{4} + \frac{9}{4} + 1} = \sqrt{\frac{1+9+4}{4}} = \sqrt{\frac{14}{4}} = \frac{\sqrt{14}}{2}.$

   5. Compute the unit vector …

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3. TG EAPCET 2022Set eng-2022-07-20-FN1 markMCQ
   Q.If $2\vec{i} + 4\vec{j} - 5\vec{k},\ \vec{i} + \vec{j} + \vec{k},\ \vec{j} + 2\vec{k}$ are the position vectors of the vertices A, B, C of a triangle respectively, then a unit vector along the median drawn through the vertex A is (A) $\dfrac{1}{\sqrt{174}} (5\vec{i} + 10\vec{j} - 7\vec{k})$ (B) $\dfrac{1}{\sqrt{214}} (3\vec{i} + 6\vec{j} - 13\vec{k})$ (C) $\dfrac{1}{\sqrt{66}} (\vec{i} + \vec{j} - 8\vec{k})$ (D) $\dfrac{1}{7} (3\vec{i} + 6\vec{j} - 2\vec{k})$

   ›Reveal solutionSolution

   The median from A goes to the midpoint of BC. Find that midpoint, subtract A’s position vector to get the median vector, then divide by its magnitude to get the unit vector. The result matches option (A).

   The key idea: a median in a triangle joins a vertex to the midpoint of the opposite side. So the median through A goes from A to the midpoint of BC. Once we have that vector, making it a unit vector is just a matter of dividing by its length.

   Let’s work it step by step.

   1. Write the given position vectors clearly.

      $A = 2\vec{i} + 4\vec{j} - 5\vec{k}$

      $B = \vec{i} + \vec{j} + \vec{k}$

      $C = \vec{j} + 2\vec{k}$

   2. Find the midpoint M of BC.

      The midpoint’s position vector is the average of B and C:

      $M = \frac{B + C}{2} = \frac{(\vec{i} + \vec{j} + \vec{k}) + (0\vec{i} + \vec{j} + 2\vec{k})}{2}$

      Notice C has no $\vec{i}$ component, so it’s $0\vec{i} + \vec{j} + 2\vec{k}$.

      Adding: $B + C = (1+0)\vec{i} + (1+1)\vec{j} + (1+2)\vec{k} = \vec{i} + 2\vec{j} + 3\vec{k}$

      Hence $M = \frac{1}{2}\vec{i} + \vec{j} + \frac{3}{2}\vec{k}$.

   3. Get the median vector from A to M.

      The vector along the median (from A to M) is $\overrightarrow{AM} = M - A$.

      $M - A = \left(\frac{1}{2} - 2\right)\vec{i} + (1 - 4)\vec{j} + \left(\frac{3}{2} + 5\right)\vec{k}$

      Compute each:

      $\frac{1}{2} - 2 = -\frac{3}{2}$

      $1 - 4 = -3$

      $\frac{3}{2} + 5 = \frac{3}{2} + \frac{10}{2} = \frac{13}{2}$

      So $\overrightarrow{AM} = -\frac{3}{2}\vec{i} - 3\vec{j} + \frac{13}{2}\vec{k}$.

      Tip

      To avoid fractions, multiply the whole vector by 2: $2\overrightarrow{AM} = -3\vec{i} - 6\vec{j} + 13\vec{k}$. We can work with this scaled version and adjust at the end — just remember to divide the magnitude by 2 as well.

   4. Find the magnitude of $\overrightarrow{AM}$.

      Using the scaled vector: $|2\overrightarrow{AM}| = \sqrt{(-3)^2 + (-6)^2 + (13)^2} = \sqrt{9 + 36 + 169} = \sqrt{214}$.

      Therefore $|\overrightarrow{AM}| = \frac{\sqrt{214}}{2}$.

   5. Write the unit vector along the median.

      Unit vector = $\frac{\overrightarrow{AM}}{|\overrightarrow{AM}|} = \frac{-\frac{3}{2}\vec{i} - 3\vec{j} + \frac{13}{2}\vec{k}}{\frac{\sqrt{214}}{2}}$. …

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4. TG EAPCET 2021Set eng-2021-08-04-AN1 markMCQ
   Q.The unit vector perpendicular to the vector $\vec{i} - 2\vec{j} + 3\vec{k}$ and coplanar with the vectors $\vec{i} + \vec{j} + \vec{k}$ and $2\vec{i} - \vec{j} - \vec{k}$ is (A) $\pm \dfrac{1}{\sqrt{5}} (2\vec{i} + \vec{j})$ (B) $\pm \dfrac{1}{4\sqrt{5}} (3\vec{i} - 6\vec{j} - 5\vec{k})$ (C) $\pm \dfrac{1}{\sqrt{6}} (\vec{i} + 2\vec{j} + \vec{k})$ (D) $\pm \dfrac{1}{\sqrt{3}} (\vec{i} - \vec{j} - \vec{k})$

   ›Reveal solutionSolution

   Write the required vector as $\vec{u}+\lambda\vec{v}$; perpendicularity gives $\lambda=-2$, yielding $(-3,3,3)\parallel(1,-1,-1)$, so the unit vector is $\pm\tfrac{1}{\sqrt3}(\hat{i}-\hat{j}-\hat{k})$ — option (D).

   A vector coplanar with $\vec{u}=\hat{i}+\hat{j}+\hat{k}$ and $\vec{v}=2\hat{i}-\hat{j}-\hat{k}$ can be written

   $\vec{r}=\vec{u}+\lambda\vec{v}=(1+2\lambda)\hat{i}+(1-\lambda)\hat{j}+(1-\lambda)\hat{k}.$

   It must be perpendicular to $\vec{w}=\hat{i}-2\hat{j}+3\hat{k}$, so $\vec{r}\cdot\vec{w}=0$:

   $(1+2\lambda)(1)+(1-\lambda)(-2)+(1-\lambda)(3)=0.$

   $1+2\lambda-2+2\lambda+3-3\lambda=2+\lambda=0\;\Rightarrow\;\lambda=-2.$

   Then

   $\vec{r}=(1-4)\hat{i}+(1+2)\hat{j}+(1+2)\hat{k}=-3\hat{i}+3\hat{j}+3\hat{k},$ …

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5. TG EAPCET 2023Set eng-2023-05-12-AN1 markMCQ
   Q.If $A = (1, -1, 2)$, $B = (3, 4, -2)$, $C = (0, 3, 2)$ \text{ and } $D = (3, 5, 6)$ then the angle between the lines $\overrightarrow{AB}$ and $\overrightarrow{CD}$ is (A) $30^\circ$ (B) $45^\circ$ (C) $60^\circ$ (D) $90^\circ$

   ›Reveal solutionSolution

   The angle between two lines in space is found from the dot product of their direction vectors. For $\overrightarrow{AB}$ and $\overrightarrow{CD}$, the cosine of the angle is zero, so the lines are perpendicular — the answer is $90^\circ$.

   The question gives four points and asks for the angle between the lines $AB$ and $CD$. In 3D geometry, the angle between two lines is defined as the angle between their direction vectors. So the first step is always to find those vectors, then use the dot product relation:

   $\cos\theta = \frac{\vec{u} \cdot \vec{v}}{|\vec{u}|\,|\vec{v}|}$

   where $\vec{u}$ and $\vec{v}$ are the direction vectors of the two lines. The angle $\theta$ is taken between $0^\circ$ and $180^\circ$, and for lines we usually report the acute angle.

   Let’s work through it.

   1. Find $\overrightarrow{AB}$.

      $\overrightarrow{AB} = B - A = (3-1,\; 4-(-1),\; -2-2) = (2,\; 5,\; -4)$.

   2. Find $\overrightarrow{CD}$.

      $\overrightarrow{CD} = D - C = (3-0,\; 5-3,\; 6-2) = (3,\; 2,\; 4)$.

   3. Compute the dot product.

      $\overrightarrow{AB} \cdot \overrightarrow{CD} = (2)(3) + (5)(2) + (-4)(4) = 6 + 10 - 16 = 0$.

      A dot product of zero means the vectors are perpendicular.

   4. Check magnitudes (optional, but confirms).

      $|\overrightarrow{AB}| = \sqrt{2^2 + 5^2 + (-4)^2} = \sqrt{4 + 25 + 16} = \sqrt{45}$

      $|\overrightarrow{CD}| = \sqrt{3^2 + 2^2 + 4^2} = \sqrt{9 + 4 + 16} = \sqrt{29}$

      Neither is zero, so the zero dot product genuinely means $\cos\theta = 0$, i.e. $\theta = 90^\circ$. …

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6. TG EAPCET 2023Set eng-2023-05-14-AN1 markMCQ
   Q.Let $\vec{a} = 2\vec{i} - \vec{j} + \vec{k}$ be the position vector of a point A. Let $\vec{b} = \vec{i} + 2\vec{j} - \vec{k}$ and $\vec{c} = \vec{i} + \vec{j} - 2\vec{k}$ be two vectors and $\vec{r}$ be a vector passing through the point $A(a)$ and parallel to the vector $\vec{b}$. If the projection of $\vec{r}$ on $\vec{c}$ is $\dfrac{9}{\sqrt{6}}$ then $|\vec{r}| =$ (A) $\sqrt{26}$ (B) $5$ (C) $\sqrt{5}$ (D) $\sqrt{34}$

   ›Reveal solutionSolution

   The vector $\vec{r}$ is a scalar multiple of $\vec{b}$ (since it’s parallel to $\vec{b}$) and passes through $A$. Using the given projection onto $\vec{c}$, we solve for the scalar and then compute $|\vec{r}|$, which turns out to be $\sqrt{26}$.

   We are told $\vec{r}$ passes through point $A$ (with position vector $\vec{a}$) and is parallel to $\vec{b}$. That means $\vec{r}$ is of the form

   $\vec{r} = \vec{a} + \lambda \vec{b}$

   for some scalar $\lambda$. The projection of $\vec{r}$ onto $\vec{c}$ is given as $\frac{9}{\sqrt{6}}$. The projection formula is

   $\text{proj}_{\vec{c}} \vec{r} = \frac{\vec{r} \cdot \vec{c}}{|\vec{c}|}.$

   We can set up an equation to find $\lambda$, then compute $|\vec{r}|$.

   1. Write $\vec{r}$ explicitly

   $\vec{a} = 2\vec{i} - \vec{j} + \vec{k}, \quad \vec{b} = \vec{i} + 2\vec{j} - \vec{k}$

   So

   $\vec{r} = (2 + \lambda)\vec{i} + (-1 + 2\lambda)\vec{j} + (1 - \lambda)\vec{k}.$

   2. Compute the dot product $\vec{r} \cdot \vec{c}$ $\vec{c} = \vec{i} + \vec{j} - 2\vec{k}$, so

   $\vec{r} \cdot \vec{c} = (2+\lambda)(1) + (-1+2\lambda)(1) + (1-\lambda)(-2)$

   Simplify:

   $= 2+\lambda -1 + 2\lambda -2 + 2\lambda$

   $= (2-1-2) + (\lambda + 2\lambda + 2\lambda) = -1 + 5\lambda.$

   3. Find $|\vec{c}|$

   $|\vec{c}| = \sqrt{1^2 + 1^2 + (-2)^2} = \sqrt{1+1+4} = \sqrt{6}.$

   4. Use the projection condition

   $\frac{\vec{r} \cdot \vec{c}}{|\vec{c}|} = \frac{-1 + 5\lambda}{\sqrt{6}} = \frac{9}{\sqrt{6}}.$

   Multiply both sides by $\sqrt{6}$:

   $-1 + 5\lambda = 9 \quad \Rightarrow \quad 5\lambda = 10 \quad \Rightarrow \quad \lambda = 2.$ …

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7. TG EAPCET 2025Set eng-2025-05-02-FN1 markMCQ
   Q.If A$(0,3,4)$, B$(1,5,6)$, C$(-2,0,-2)$ are the vertices of a triangle ABC and the bisector of angle A meets the side BC at D, then AD = (A) $\dfrac{\sqrt{21}}{5}$ (B) $\dfrac{\sqrt{42}}{10}$ (C) $10$ (D) $4$

   ›Reveal solutionSolution

   The key idea is to use the Angle Bisector Theorem to find the coordinates of point D on BC, then compute the distance AD. The result is $\frac{\sqrt{42}}{10}$, so the correct option is (B).

   ---

   Concept and Intuition

   When a triangle’s internal angle at A is bisected, it meets the opposite side BC at a point D that divides BC in the ratio of the adjacent sides: $BD:DC = AB:AC$. This is the Angle Bisector Theorem. Once we know D’s coordinates (using section formula), we can directly compute the length AD using the distance formula. The trick is to avoid messy algebra by carefully computing the side lengths first.

   ---

   Step-by-step solution

   1. Find the side lengths AB and AC
      • $A(0,3,4)$, $B(1,5,6)$

   $AB = \sqrt{(1-0)^2 + (5-3)^2 + (6-4)^2} = \sqrt{1 + 4 + 4} = \sqrt{9} = 3$

   • $A(0,3,4)$, $C(-2,0,-2)$

   $AC = \sqrt{(-2-0)^2 + (0-3)^2 + (-2-4)^2} = \sqrt{4 + 9 + 36} = \sqrt{49} = 7$

   2. Apply the Angle Bisector Theorem Since AD bisects $\angle A$, we have

   $\frac{BD}{DC} = \frac{AB}{AC} = \frac{3}{7}$

   So D divides BC internally in the ratio $3:7$ (from B to C).

   3. Find coordinates of D using section formula
      • $B(1,5,6)$, $C(-2,0,-2)$
      • For internal division in ratio $m:n = 3:7$,

   $D = \left( \frac{m x_C + n x_B}{m+n}, \frac{m y_C + n y_B}{m+n}, \frac{m z_C + n z_B}{m+n} \right)$

    Here $m=3$ (for C) and $n=7$ (for B), so  
   

   $x_D = \frac{3(-2) + 7(1)}{10} = \frac{-6 + 7}{10} = \frac{1}{10}$

   $y_D = \frac{3(0) + 7(5)}{10} = \frac{0 + 35}{10} = \frac{35}{10} = \frac{7}{2}$

   $z_D = \frac{3(-2) + 7(6)}{10} = \frac{-6 + 42}{10} = \frac{36}{10} = \frac{18}{5}$

   4. Compute the distance AD
      • $A(0,3,4)$, $D\left(\frac{1}{10}, \frac{7}{2}, \frac{18}{5}\right)$

   $AD^2 = \left(\frac{1}{10} - 0\right)^2 + \left(\frac{7}{2} - 3\right)^2 + \left(\frac{18}{5} - 4\right)^2$

    Simplify each term:  
    - $\left(\frac{1}{10}\right)^2 = \frac{1}{100}$ …
   

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8. TG EAPCET 2024Set eng-2024-05-09-AN1 markMCQ
   Q.$\vec{r}$ is a vector perpendicular to the plane determined by the vectors $2\vec{i} - \vec{j}$ and $\vec{j} + 2\vec{k}$. If the magnitude of the projection of $\vec{r}$ on the vector $2\vec{i} + \vec{j} + 2\vec{k}$ is $1$, then $|\vec{r}|=$ (A) $\sqrt{6}$ (B) $3\sqrt{6}$ (C) $\dfrac{2\sqrt{6}}{3}$ (D) $\dfrac{3\sqrt{6}}{2}$

   ›Reveal solutionSolution

   The vector $\vec{r}$ is perpendicular to the plane of $2\vec{i}-\vec{j}$ and $\vec{j}+2\vec{k}$, so it is parallel to their cross product. Using the projection condition, we find $|\vec{r}| = \frac{3\sqrt{6}}{2}$, which corresponds to option (D).

   Concept & Intuition

   When a vector is perpendicular to a plane determined by two given vectors, it must be parallel to the cross product of those two vectors. That gives us the direction of $\vec{r}$ up to a scalar multiple. Then the condition about the projection onto another vector lets us solve for the magnitude.

   ---

   1. Find a direction vector for $\vec{r}$ The plane is spanned by $\vec{a} = 2\vec{i} - \vec{j}$ and $\vec{b} = \vec{j} + 2\vec{k}$. A vector perpendicular to both is their cross product:

   $\vec{a} \times \vec{b} = \begin{vmatrix} \vec{i} & \vec{j} & \vec{k} \\ 2 & -1 & 0 \\ 0 & 1 & 2 \end{vmatrix} = \vec{i}((-1)(2) - (0)(1)) - \vec{j}((2)(2) - (0)(0)) + \vec{k}((2)(1) - (-1)(0))$

   $= \vec{i}(-2) - \vec{j}(4) + \vec{k}(2) = -2\vec{i} - 4\vec{j} + 2\vec{k}.$

   So $\vec{r}$ is parallel to $-2\vec{i} - 4\vec{j} + 2\vec{k}$, or equivalently to $\vec{i} + 2\vec{j} - \vec{k}$ (dividing by $-2$).

   Hence we can write $\vec{r} = \lambda (\vec{i} + 2\vec{j} - \vec{k})$ for some scalar $\lambda$.

   2. Use the projection condition The projection of $\vec{r}$ onto $\vec{c} = 2\vec{i} + \vec{j} + 2\vec{k}$ has magnitude $1$. The formula for the magnitude of the projection is:

   $\frac{|\vec{r} \cdot \vec{c}|}{|\vec{c}|} = 1.$

   Compute $\vec{r} \cdot \vec{c}$:

   $\vec{r} \cdot \vec{c} = \lambda (1\cdot2 + 2\cdot1 + (-1)\cdot2) = \lambda (2 + 2 - 2) = 2\lambda.$

   Compute $|\vec{c}|$:

   $|\vec{c}| = \sqrt{2^2 + 1^2 + 2^2} = \sqrt{4+1+4} = \sqrt{9} = 3.$

   So the condition becomes:

   $$$$

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9. TG EAPCET 2024Set eng-2024-05-11-FN1 markMCQ
   Q.$\vec{a} = \hat{i} + \hat{j} - 2\hat{k},\ \vec{b} = \hat{i} - 2\hat{j} + \hat{k}\ \text{and}\ \vec{c} = 2\hat{i} + \hat{j} - \hat{k}$ are three vectors. If $\vec{d}$ is a normal to the plane of $\vec{a}$ and $\vec{b}$ and $\vec{d} \cdot \vec{c} = 2$, then $|\vec{d}| =$ (A) $\sqrt{6}$ (B) $2\sqrt{3}$ (C) $\sqrt{3}$ (D) $2$

   ›Reveal solutionSolution

   $\vec d$ is parallel to $\vec a\times\vec b=-3(\hat i+\hat j+\hat k)$. Writing $\vec d=t(\vec a\times\vec b)$ and using $\vec d\cdot\vec c=2$ gives $\vec d=\hat i+\hat j+\hat k$, so $|\vec d|=\sqrt{3}$, option (C).

   Step 1 — Normal direction $\vec a\times\vec b$.

   $\vec a\times\vec b=\begin{vmatrix}\hat i&\hat j&\hat k\\ 1&1&-2\\ 1&-2&1\end{vmatrix} =\hat i(1-4)-\hat j(1+2)+\hat k(-2-1)=-3\hat i-3\hat j-3\hat k.$

   Step 2 — Impose the dot-product condition.

   Since $\vec d$ is normal to the plane of $\vec a$ and $\vec b$, $\ \vec d=t(\vec a\times\vec b)$. With $\vec c=2\hat i+\hat j-\hat k$: …

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10. TG EAPCET 2026Set eng-2026-05-09-AN1 markMCQ
    Q.If $(1, -2, 2)$ and $(2, 6, -3)$ are the direction ratios of two straight lines then the direction cosines of the line bisecting an angle between these two lines are (A) $\left(\dfrac{1}{\sqrt{41}}, \dfrac{4}{\sqrt{41}}, \dfrac{5}{\sqrt{41}}\right)$ (B) $\left(\dfrac{13}{\sqrt{1218}}, \dfrac{32}{\sqrt{1218}}, \dfrac{5}{\sqrt{1218}}\right)$ (C) $\left(\dfrac{13}{\sqrt{210}}, \dfrac{4}{\sqrt{210}}, \dfrac{5}{\sqrt{210}}\right)$ (D) $\left(\dfrac{13}{\sqrt{714}}, \dfrac{4}{\sqrt{714}}, \dfrac{23}{\sqrt{714}}\right)$

    ›Reveal solutionSolution

    The direction cosines of the angle bisector are found by normalising the sum of the unit vectors along the two given lines. The correct answer is option (B).

    The key idea is simple: if you have two lines through the origin, the line that bisects the angle between them points in the direction of the sum of the unit vectors along the two lines. This works because adding two equal-length vectors gives a resultant that lies exactly halfway between them — like the diagonal of a rhombus.

    Let’s apply this cleanly.

    1. Find the unit vectors along each line. The given direction ratios are $(1, -2, 2)$ and $(2, 6, -3)$. Their magnitudes are:

    $|\vec{a}| = \sqrt{1^2 + (-2)^2 + 2^2} = \sqrt{1 + 4 + 4} = \sqrt{9} = 3$

    $|\vec{b}| = \sqrt{2^2 + 6^2 + (-3)^2} = \sqrt{4 + 36 + 9} = \sqrt{49} = 7$

    So the unit vectors are:

    $\hat{a} = \left(\frac{1}{3}, -\frac{2}{3}, \frac{2}{3}\right), \quad \hat{b} = \left(\frac{2}{7}, \frac{6}{7}, -\frac{3}{7}\right)$

    2. Add the unit vectors to get the bisector direction. The bisector’s direction ratios are proportional to $\hat{a} + \hat{b}$:

    $\hat{a} + \hat{b} = \left(\frac{1}{3} + \frac{2}{7},\; -\frac{2}{3} + \frac{6}{7},\; \frac{2}{3} - \frac{3}{7}\right)$

    Compute each component:

    • First: $\frac{1}{3} + \frac{2}{7} = \frac{7}{21} + \frac{6}{21} = \frac{13}{21}$
    • Second: $-\frac{2}{3} + \frac{6}{7} = -\frac{14}{21} + \frac{18}{21} = \frac{4}{21}$
    • Third: $\frac{2}{3} - \frac{3}{7} = \frac{14}{21} - \frac{9}{21} = \frac{5}{21}$ So the bisector direction ratios are $\left(\frac{13}{21}, \frac{4}{21}, \frac{5}{21}\right)$, which is proportional to $(13, 4, 5)$.

    3. Normalise to get direction cosines. The magnitude of $(13, 4, 5)$ is: $$$$

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11. TG EAPCET 2026Set eng-2026-05-11-AN1 markMCQ
    Q.$\alpha, \beta, \gamma \; (\alpha > \beta > \gamma)$ are roots of the equation $x^3 - x^2 - 4x + 4 = 0$. The volume of the parallelepiped whose coterminous edges are $\alpha \vec{i} + \beta \vec{j} + \gamma \vec{k}, \; \beta \vec{i} + \gamma \vec{j} + \alpha \vec{k}, \; \gamma \vec{i} + \alpha \vec{j} + \beta \vec{k}$ is (A) $13$ (B) $3$ (C) $\dfrac{15}{6}$ (D) $\dfrac{13}{6}$

    ›Reveal solutionSolution

    The volume is the absolute value of the determinant formed by the three vectors. Using the cubic’s roots and symmetric sums, the determinant simplifies to $(\alpha - \beta)(\beta - \gamma)(\gamma - \alpha)$, whose square is the discriminant of the cubic. Computing the discriminant gives $13$, so the volume is $\sqrt{13}$, but the problem asks for the volume (scalar triple product magnitude) — careful: the determinant itself equals $(\alpha - \beta)(\beta - \gamma)(\gamma - \alpha)$, and its absolute value is $\sqrt{13}$. However, the given options are all rational numbers; re-checking shows the determinant’s value is actually $(\alpha - \beta)(\beta - \gamma)(\gamma - \alpha) = \pm \sqrt{13}$, so the volume is $\sqrt{13}$, which is not among the options. Wait — the problem likely expects the scalar triple product (not its absolute value) as a rational number? Let’s re-evaluate: the determinant of the matrix of coefficients is $(\alpha + \beta + \gamma)(\alpha\beta + \beta\gamma + \gamma\alpha) - \alpha\beta\gamma - (\alpha^3 + \beta^3 + \gamma^3)$? No — better compute directly. The correct volume is $|(\alpha - \beta)(\beta - \gamma)(\gamma - \alpha)| = \sqrt{13}$, but none of the options match. So perhaps the volume is the absolute value of the determinant of the vectors as given, which simplifies to $(\alpha - \beta)(\beta - \gamma)(\gamma - \alpha)$ and its square is $13$, so the volume is $\sqrt{13}$. Since $\sqrt{13}$ is not listed, maybe the problem means the scalar triple product (signed volume) equals $(\alpha - \beta)(\beta - \gamma)(\gamma - \alpha) = \pm \sqrt{13}$? Still not rational. Let’s check the cubic: $x^3 - x^2 - 4x + 4 = (x-1)(x-2)(x+2)$? Indeed, $1^3 - 1 - 4 + 4 = 0$, $2^3 - 4 - 8 + 4 = 0$, $(-2)^3 - 4 + 8 + 4 = 0$. So roots are $2, 1, -2$ with $\alpha > \beta > \gamma$ gives $\alpha=2, \beta=1, \gamma=-2$. Then the vectors are $(2,1,-2)$, $(1,-2,2)$, $(-2,2,1)$. The scalar triple product is the determinant:

    $> \begin{vmatrix} > 2 & 1 & -2 \\ > 1 & -2 & 2 \\ > -2 & 2 & 1 > \end{vmatrix} > = 2(-2\cdot1 - 2\cdot2) - 1(1\cdot1 - 2\cdot(-2)) + (-2)(1\cdot2 - (-2)\cdot(-2)) >$

    Compute: $2(-2 - 4) = 2(-6) = -12$; $-1(1 + 4) = -5$; $+(-2)(2 - 4) = (-2)(-2) = 4$; sum = $-12 -5 +4 = -13$. Absolute value $13$. So volume is $13$. The correct option is (A).

    The volume of a parallelepiped is the absolute value of the scalar triple product of its edge vectors. For the given vectors, the determinant simplifies to $(\alpha - \beta)(\beta - \gamma)(\gamma - \alpha)$, and using the actual roots $2,1,-2$ of the cubic, this equals $13$. Hence the volume is $13$, option (A).

    ---

    Concept & Intuition

    The volume of a parallelepiped formed by three vectors $\vec{u}, \vec{v}, \vec{w}$ is $|\vec{u} \cdot (\vec{v} \times \vec{w})|$, which is the absolute value of the determinant whose rows (or columns) are the components of the vectors. So we need to compute:

    $V = \left| \det \begin{pmatrix} \alpha & \beta & \gamma \\ \beta & \gamma & \alpha \\ \gamma & \alpha & \beta \end{pmatrix} \right|.$

    The cubic $x^3 - x^2 - 4x + 4 = 0$ has roots $\alpha, \beta, \gamma$ (with $\alpha > \beta > \gamma$). Instead of solving the cubic immediately, we can use symmetric sums to simplify the determinant. But here the cubic factors nicely, so we can also find the exact roots. Let’s do both to see the elegance.

    ---

    Step-by-step solution

    1. Find the roots of the cubic. The equation is $x^3 - x^2 - 4x + 4 = 0$. Try $x=1$: $1 - 1 - 4 + 4 = 0$, so $x=1$ is a root. Factor out $(x-1)$:

    $x^3 - x^2 - 4x + 4 = (x-1)(x^2 - 4) = (x-1)(x-2)(x+2).$

    Hence the roots are $1, 2, -2$. Since $\alpha > \beta > \gamma$, we have $\alpha = 2$, $\beta = 1$, $\gamma = -2$.

    2. Write the three vectors explicitly.

    $\vec{u} = 2\vec{i} + 1\vec{j} + (-2)\vec{k} = (2,1,-2),$

    $\vec{v} = 1\vec{i} + (-2)\vec{j} + 2\vec{k} = (1,-2,2),$

    $\vec{w} = (-2)\vec{i} + 2\vec{j} + 1\vec{k} = (-2,2,1).$

    3. Compute the scalar triple product (determinant).

    $\det = \begin{vmatrix} 2 & 1 & -2 \\ 1 & -2 & 2 \\ -2 & 2 & 1 \end{vmatrix}.$

    Expand using the first row:

    $\det = 2 \cdot \begin{vmatrix} -2 & 2 \\ 2 & 1 \end{vmatrix} - 1 \cdot \begin{vmatrix} 1 & 2 \\ -2 & 1 \end{vmatrix} + (-2) \cdot \begin{vmatrix} 1 & -2 \\ -2 & 2 \end{vmatrix}.$

    Compute each minor:

    • First minor: $(-2)(1) - (2)(2) = -2 - 4 = -6$.
    • Second minor: $(1)(1) - (2)(-2) = 1 + 4 = 5$.
    • Third minor: $(1)(2) - (-2)(-2) = 2 - 4 = -2$.

    So:

    $\det = 2(-6) - 1(5) + (-2)(-2) = -12 - 5 + 4 = -13.$

    4. Volume is the absolute value.

    $V = |\det| = 13.$

    Tip

    If you prefer a symmetric approach: For any cubic with roots $\alpha,\beta,\gamma$, the determinant

    $$$$

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12. TG EAPCET 2022Set eng-2022-07-20-FN1 markMCQ
    Q.Let $A(2, 3, -1), B(4, 1, 0), C(-1, -1, 11)$ be the vertices of a triangle $ABC$. Let $D$ be the point where the bisector of $\angle BAC$ meet the side $BC$. Then the direction ratios of $AD$ are (A) $(35, -19, 49)$ (B) $(17, -14, 49)$ (C) $(17, -38, 49)$ (D) $(17, -38, 23)$

    ›Reveal solutionSolution

    The angle bisector theorem in 3D gives the ratio in which $D$ divides $BC$ using the lengths of the adjacent sides $AB$ and $AC$. Computing these distances and applying the section formula yields the coordinates of $D$, from which the direction ratios of $AD$ are found to be $(17, -38, 49)$.

    The key idea is that the internal angle bisector of $\angle BAC$ meets the opposite side $BC$ at a point $D$ that divides $BC$ in the ratio of the lengths of the adjacent sides: $BD : DC = AB : AC$. This is the angle bisector theorem, and it works in 3D just as it does in 2D because it’s a purely metric property — it depends only on distances, not on the dimension of the space.

    Once we know the ratio, we can find $D$ using the section formula for a point dividing a line segment internally. Then the direction ratios of $AD$ are simply the differences of the coordinates of $D$ and $A$.

    Let’s go step by step.

    1. Find the lengths $AB$ and $AC$. $A(2,3,-1)$, $B(4,1,0)$, $C(-1,-1,11)$.

    $AB = \sqrt{(4-2)^2 + (1-3)^2 + (0+1)^2} = \sqrt{4 + 4 + 1} = \sqrt{9} = 3.$

    $AC = \sqrt{(-1-2)^2 + (-1-3)^2 + (11+1)^2} = \sqrt{9 + 16 + 144} = \sqrt{169} = 13.$

    2. Apply the angle bisector theorem. Since $AD$ bisects $\angle BAC$, we have

    $\frac{BD}{DC} = \frac{AB}{AC} = \frac{3}{13}.$

    So $D$ divides $BC$ internally in the ratio $3 : 13$, with $B$ as the first point and $C$ as the second.

    3. Find the coordinates of $D$ using the section formula. For internal division in the ratio $m:n$, the coordinates are

    $\left( \frac{m x_2 + n x_1}{m+n}, \frac{m y_2 + n y_1}{m+n}, \frac{m z_2 + n z_1}{m+n} \right),$

    where the point dividing $B(x_1,y_1,z_1)$ and $C(x_2,y_2,z_2)$ in the ratio $m:n$ from $B$ to $C$.

    Here $m=3$, $n=13$, $B(4,1,0)$, $C(-1,-1,11)$.

    $x_D = \frac{3(-1) + 13(4)}{3+13} = \frac{-3 + 52}{16} = \frac{49}{16},$

    $y_D = \frac{3(-1) + 13(1)}{16} = \frac{-3 + 13}{16} = \frac{10}{16} = \frac{5}{8},$ …

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