Q.If $a = i + 2j - 3k$ and $b = 3i - j + 2k$, then show that $a + b$ and $a - b$ are perpendicular to each other.

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1. CBSE 2026Set 65/1/11 markMCQ
   Q.The value of $p$ for which vectors $\hat{i} + 2\hat{j} + 3\hat{k}$ and $2\hat{i} - p\hat{j} + \hat{k}$ are perpendicular to each other is (A) $0$ (B) $1$ (C) $\dfrac{5}{2}$ (D) $-\dfrac{5}{2}$

   ›Reveal solutionSolution

   Two vectors are perpendicular when their dot product equals zero. Setting the dot product of $\hat{i} + 2\hat{j} + 3\hat{k}$ and $2\hat{i} - p\hat{j} + \hat{k}$ to zero gives $p = \frac{5}{2}$, which corresponds to option (C).

   The key idea here is the perpendicular vectors condition: two vectors are perpendicular (orthogonal) if and only if their dot product is zero. This is a fundamental geometric fact — the dot product measures how much one vector "projects" onto another; when they're at right angles, that projection is zero.

   Let’s apply this step by step.

   1. Write the vectors in component form.

      Let $\vec{a} = \hat{i} + 2\hat{j} + 3\hat{k}$ and $\vec{b} = 2\hat{i} - p\hat{j} + \hat{k}$.

      In component notation:

      $\vec{a} = (1, 2, 3)$ and $\vec{b} = (2, -p, 1)$.

   2. Recall the dot product formula.

      For vectors $(x_1, y_1, z_1)$ and $(x_2, y_2, z_2)$,

   $\vec{a} \cdot \vec{b} = x_1 x_2 + y_1 y_2 + z_1 z_2.$

   3. Set up the perpendicular condition. We require $\vec{a} \cdot \vec{b} = 0$. So:

   $(1)(2) + (2)(-p) + (3)(1) = 0.$

   4. Simplify the equation. …

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2. CBSE 20261 mark
   Q.Assertion (A): Lines given by $x = py + q, z = ry + s$ and $x = p'y + q', z = r'y + s'$ are perpendicular if $pp' + rr' = 1$. Reason (R): Two lines $\vec{r} = \vec{a}_1 + \lambda \vec{b}_1$ and $\vec{r} = \vec{a}_2 + \mu \vec{b}_2$ are perpendicular if $\vec{b}_1 \cdot \vec{b}_2 = 0$.

   ›Reveal solutionSolution

   The key idea is to convert the given symmetric equations into vector form, extract the direction vectors, and apply the perpendicularity condition $\vec{b}_1 \cdot \vec{b}_2 = 0$. The assertion is false because the correct condition is $pp' + rr' = -1$, not $+1$.

   Let’s understand why. The problem tests two things: first, how to read direction vectors from a pair of linear equations representing a line, and second, the precise condition for perpendicular lines in 3D.

   The Core Concept

   Two lines in space are perpendicular when their direction vectors are orthogonal — that is, their dot product is zero. The Reason (R) states this correctly: for lines $\vec{r} = \vec{a}_1 + \lambda \vec{b}_1$ and $\vec{r} = \vec{a}_2 + \mu \vec{b}_2$, perpendicularity means $\vec{b}_1 \cdot \vec{b}_2 = 0$.

   The trick lies in the Assertion (A). The given equations $x = py + q, z = ry + s$ represent a line, but not in the standard symmetric form. We need to extract its direction vector.

   Watch out

   A common mistake is to read the coefficients of $y$ directly as direction ratios. That would give $(p, 1, r)$, but this is incorrect — the equations are not in the form $\frac{x - x_0}{a} = \frac{y - y_0}{b} = \frac{z - z_0}{c}$.

   Let’s work through the extraction properly.

   Step-by-Step Solution

   1. Rewrite the line in symmetric form

   The equations $x = py + q$ and $z = ry + s$ both express $x$ and $z$ in terms of $y$. This means $y$ acts as a parameter. Let $y = t$. Then:

   • $x = p t + q$
   • $y = t$
   • $z = r t + s$

   So the parametric form is:

   $(x, y, z) = (q, 0, s) + t(p, 1, r)$

   The direction vector of the first line is $\vec{b}_1 = (p, 1, r)$.

   2. Similarly for the second line

   For $x = p'y + q', z = r'y + s'$, let $y = u$. Then:

   • $x = p' u + q'$
   • $y = u$
   • $z = r' u + s'$

   So the direction vector is $\vec{b}_2 = (p', 1, r')$.

   3. Apply the perpendicular condition …

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3. CBSE 2026Set 65/1/11 markMCQ
   Q.Assertion (A): The lines $x = py + q$, $z = ry + s$ and $x = p'y + q'$, $z = r'y + s'$ are perpendicular to each other when $pp' + rr' = 1$. Reason (R): Two lines $\vec{r} = \vec{a}_1 + \lambda\vec{b}_1$ and $\vec{r} = \vec{a}_2 + \mu\vec{b}_2$ are perpendicular to each other if $\vec{b}_1 \cdot \vec{b}_2 = 0$. (A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of the Assertion (A). (B) Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of the Assertion (A). (C) Assertion (A) is true and Reason (R) is false. (D) Assertion (A) is false and Reason (R) is true.

   ›Reveal solutionSolution

   The condition for two lines to be perpendicular is that the dot product of their direction vectors is zero. Reason (R) correctly states this. Assertion (A) provides an incorrect condition for perpendicularity based on the given line equations. Therefore, Assertion (A) is false, and Reason (R) is true.

   The core concept for determining if two lines are perpendicular in 3D space relies on their direction vectors. A line's direction vector indicates the path it follows. If two lines are perpendicular, their direction vectors must also be perpendicular. The mathematical condition for two non-zero vectors to be perpendicular is that their dot product is zero. This is a fundamental property of the dot product.

   Let's evaluate the given Assertion and Reason.

   Evaluating Reason (R)

   Reason (R) states: "Two lines $\vec{r} = \vec{a}_1 + \lambda\vec{b}_1$ and $\vec{r} = \vec{a}_2 + \mu\vec{b}_2$ are perpendicular to each other if $\vec{b}_1 \cdot \vec{b}_2 = 0$."

   The vector equation of a line is $\vec{r} = \vec{a} + \lambda\vec{b}$, where $\vec{a}$ is the position vector of a point on the line and $\vec{b}$ is the direction vector of the line.

   The vectors $\vec{b}_1$ and $\vec{b}_2$ are the direction vectors of the respective lines. If the lines are perpendicular, their direction vectors must be perpendicular. The dot product of two perpendicular vectors is indeed zero. This statement is a correct and fundamental principle in vector algebra and 3D geometry.

   Therefore, Reason (R) is true.

   Evaluating Assertion (A)

   Assertion (A) states: "The lines $x = py + q$, $z = ry + s$ and $x = p'y + q'$, $z = r'y + s'$ are perpendicular to each other when $pp' + rr' = 1$."

   To check this assertion, we first need to find the direction vectors for each line from their given equations. The equations are given in a non-standard form, so we will convert them to the symmetric form $\frac{x-x_0}{a} = \frac{y-y_0}{b} = \frac{z-z_0}{c}$, where $\langle a, b, c \rangle$ is the direction vector.

   1. Find the direction vector for the first line: The equations are $x = py + q$ and $z = ry + s$. From $x = py + q$, we can write $y = \frac{x-q}{p}$. From $z = ry + s$, we can write $y = \frac{z-s}{r}$. Combining these, we get the symmetric form:

   $\frac{x-q}{p} = \frac{y-0}{1} = \frac{z-s}{r}$

   The direction vector for the first line, $\vec{b}_1$, is $\langle p, 1, r \rangle$.
   

   2. Find the direction vector for the second line:

   The equations are $x = p'y + q'$ and $z = r'y + s'$.

   Similarly, from $x = p'y + q'$, we get $y = \frac{x-q'}{p'}$.

   From $z = r'y + s'$, we get $y = \frac{z-s'}{r'}$.

   Combining these, we get the symmetric form:

   $\frac{x-q'}{p'} = \frac{y-0}{1} = \frac{z-s'}{r'}$

   The direction vector for the second line, $\vec{b}_2$, is $\langle p', 1, r' \rangle$.
   

   3. Apply the perpendicularity condition:

   According to Reason (R), for the two lines to be perpendicular, the dot product of their direction vectors must be zero: $\vec{b}_1 \cdot \vec{b}_2 = 0$. …

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4. CBSE 2026Set A1 markMCQ
   Q.If $3\vec{i} + \vec{j} - 2\vec{k}$ and $\vec{i} + \lambda\vec{j} - 3\vec{k}$ are perpendicular to each other then the value of $\lambda =$

   (a) $-3$

   (b) $-6$

   (c) $-9$

   (d) $-1$

   ›Reveal solutionSolution

   Set the dot product to zero.

   Perpendicular $\Rightarrow$ dot product $= 0$: …

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5. CBSE 2026Set A1 markMCQ
   Q.If $|\vec{a} + \vec{b}| = |\vec{a} - \vec{b}|$ then

   (a) $|\vec{a}| = |\vec{b}|$

   (b) $\vec{a} \parallel \vec{b}$

   (c) $\vec{a} \perp \vec{b}$

   (d) none of these

   ›Reveal solutionSolution

   $|\vec{a}+\vec{b}|=|\vec{a}-\vec{b}|$ means the vectors are perpendicular.

   Square both sides:

   $|\vec{a}+\vec{b}|^2 = |\vec{a}-\vec{b}|^2$

   $|\vec{a}|^2 + 2\vec{a}\cdot\vec{b} + |\vec{b}|^2 = |\vec{a}|^2 - 2\vec{a}\cdot\vec{b} + |\vec{b}|^2.$ …

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6. CBSE 2026Set A1 markMCQ
   Q.If two planes $x - 4y + \lambda z + 3 = 0$ and $2x + 2y + 3z = 5$ are perpendicular to each other then $\lambda =$

   (a) $1$

   (b) $2$

   (c) $3$

   (d) $4$

   ›Reveal solutionSolution

   Normals must be perpendicular: their dot product is zero.

   Normals are $(1,-4,\lambda)$ and $(2,2,3)$. For the planes to be perpendicular: …

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7. CBSE 2026Set A1 markMCQ
   Q.If the line $\frac{x-x_1}{a_1} = \frac{y-y_1}{b_1} = \frac{z-z_1}{c_1}$ is parallel to the plane $a_2 x + b_2 y + c_2 z + d = 0$ then

   (a) $\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}$

   (b) $a_1 x + b_1 y + c_1 z + d = 0$

   (c) $a_1 a_2 + b_1 b_2 + c_1 c_2 = 0$

   (d) none of these

   ›Reveal solutionSolution

   Line $\parallel$ plane $\Rightarrow$ line direction $\perp$ plane normal.

   The line has direction ratios $(a_1,b_1,c_1)$ and the plane has normal $(a_2,b_2,c_2)$. If the line is parallel to the plane, its direction lies in the plane, hence is perpendicular to the normal. S …

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8. CBSE 2026Set ANNUAL1 markMCQ
   Q.If the straight lines $\frac{x+1}{1}=\frac{y+2}{\lambda}=\frac{z-1}{-1}$ and $\frac{x-1}{-\lambda}=\frac{y+1}{2}=\frac{z+1}{1}$ are perpendicular to each other, then the value of $\lambda$ is

   (a) $0$

   (b) $1$

   (c) $2$

   (d) $3$

   ›Reveal solutionSolution

   Two lines are perpendicular when the dot product of their direction ratios is zero.

   Direction ratios: line 1 = $(1,\lambda,-1)$, line 2 = $(-\lambda,2,1)$.

   …

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9. CBSE 2026Set ANNUAL1 markMCQ
   Q.For what value of $x$, vectors $x\hat{i}-3\hat{j}-5\hat{k}$ and $-\hat{i}+\hat{j}+2\hat{k}$ are perpendicular to each other?

   (a) 4

   (b) 7

   (c) -13

   (d) None of these

   ›Reveal solutionSolution

   Two vectors are perpendicular exactly when their dot product is zero.

   Let $\vec u = x\hat i - 3\hat j - 5\hat k$ and $\vec v = -\hat i+\hat j+2\hat k$.

   …

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10. CBSE 2026Set ANNUAL1 markMCQ
    Q.If a⃗ and b⃗ are two vectors such that |a⃗| = 2 and |b⃗| = 5, then the value of λ for which a⃗ + λb⃗ and a⃗ − λb⃗ will be perpendicular is ................. .

    (a) 1/2

    (b) 2/3

    (c) 2/5

    (d) 3/4

    ›Reveal solutionSolution

    Perpendicularity means the dot product of $\vec a+\lambda\vec b$ and $\vec a-\lambda\vec b$ is zero.

    $(\vec a+\lambda\vec b)\cdot(\vec a-\lambda\vec b) = |\vec a|^2 - \lambda^2|\vec b|^2 = 0$

    …

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11. CBSE 2026Set ANNUAL1 markMCQ
    Q.The two lines $x=ay+b,\ z=cy+d$ and $x=a_{1}y+b_{1},\ z=c_{1}y+d_{1}$ are perpendicular to each other, if

    (a) $\dfrac{a}{a_{1}}+\dfrac{c}{c_{1}}=1$

    (b) $\dfrac{a}{a_{1}}+\dfrac{c}{c_{1}}=-1$

    (c) $aa_{1}+cc_{1}=1$

    (d) $aa_{1}+cc_{1}=-1$

    ›Reveal solutionSolution

    Direction ratios are $(a,1,c)$ and $(a_1,1,c_1)$; perpendicularity gives $aa_1+1+cc_1=0$, i.e. $aa_1+cc_1=-1$.

    For the first line, taking $y=t$: $x=at+b,\ y=t,\ z=ct+d$, so its direction ratios are $(a,1,c)$.

    Similarly the second line has direction ratios $(a_{1},1,c_{1})$.

    …

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12. CBSE 2025Set E1 markMCQ
    Q.If the direction ratios of two mutually perpendicular lines are $5, 2, 4$ and $4, 8, x$ then the value $x$ is

    (a) $9$

    (b) $-9$

    (c) $8$

    (d) $-8$

    ›Reveal solutionSolution

    Perpendicular lines: $a_1a_2+b_1b_2+c_1c_2=0\Rightarrow 20+16+4x=0\Rightarrow x=-9$.

    Two lines are perpendicular when the sum of the products of corresponding direction ratios is zero:

    …

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