Q.(a) A vector $\vec{a}$ makes equal angles with all the three axes. If the magnitude of the vector is $5\sqrt{3}$ units, find $\vec{a}$.

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1. CBSE 2026Set 65/2/11 markMCQ
   Q.Direction cosines of the line given by equations $\frac{2x-1}{4} = \frac{1-y}{3} = \frac{-z}{6}$ are (A) $2, -3, -6$ (B) $\frac{2}{7}, \frac{-3}{7}, \frac{-6}{7}$ (C) $\frac{2}{7}, \frac{-3}{7}, \frac{6}{7}$ (D) $\frac{4}{\sqrt{61}}, \frac{-3}{\sqrt{61}}, \frac{-6}{\sqrt{61}}$

   ›Reveal solutionSolution

   To find direction cosines, first convert the line's equation to the standard symmetric form $\frac{x-x_1}{a} = \frac{y-y_1}{b} = \frac{z-z_1}{c}$. The denominators $(a, b, c)$ are the direction ratios. Normalize these ratios by dividing by their magnitude $\sqrt{a^2+b^2+c^2}$ to get the direction cosines. The direction cosines are $\boxed{\frac{2}{7}, \frac{-3}{7}, \frac{-6}{7}}$.

   Concept and Intuition

   A line in 3D space has a specific orientation, which can be described by its direction. This direction is represented by a vector parallel to the line.

   Direction Ratios: If a vector $\vec{d} = a\hat{i} + b\hat{j} + c\hat{k}$ is parallel to a line, then the numbers $(a, b, c)$ are called the direction ratios of the line. There are infinitely many sets of direction ratios for a given line (e.g., $(2a, 2b, 2c)$ would also be direction ratios).

   Direction Cosines: These are a unique set of direction ratios that are normalized. If $(a, b, c)$ are direction ratios, then the direction cosines $(l, m, n)$ are given by:

   $l = \frac{a}{\sqrt{a^2+b^2+c^2}}$

   $m = \frac{b}{\sqrt{a^2+b^2+c^2}}$

   $n = \frac{c}{\sqrt{a^2+b^2+c^2}}$

   The direction cosines are essentially the components of a unit vector parallel to the line. They are the cosines of the angles the line makes with the positive $x, y, z$ axes, respectively. An important property is that $l^2 + m^2 + n^2 = 1$.

   The standard symmetric form of the equation of a line passing through a point $(x_1, y_1, z_1)$ and having direction ratios $(a, b, c)$ is:

   $\frac{x-x_1}{a} = \frac{y-y_1}{b} = \frac{z-z_1}{c}$

   The key insight here is that for the denominators to represent the direction ratios, the numerators must be in the form $(x-x_1)$, $(y-y_1)$, and $(z-z_1)$. If they are not, we must algebraically manipulate the equation to achieve this form first.

   Step-by-Step Solution

   1. Convert the given equation to standard symmetric form.

      The given equation is $\frac{2x-1}{4} = \frac{1-y}{3} = \frac{-z}{6}$.

      We need to transform each part so that the numerators are of the form $(x-x_1)$, $(y-y_1)$, and $(z-z_1)$.

      • For the first part, $\frac{2x-1}{4}$:

        Factor out 2 from the numerator: $\frac{2(x - 1/2)}{4}$.

        Simplify: $\frac{x - 1/2}{2}$.

      • For the second part, $\frac{1-y}{3}$:

        Factor out -1 from the numerator: $\frac{-(y - 1)}{3}$.

        Move the negative sign to the denominator: $\frac{y - 1}{-3}$.

      • For the third part, $\frac{-z}{6}$:

        Factor out -1 from the numerator: $\frac{-(z - 0)}{6}$.

        Move the negative sign to the denominator: $\frac{z - 0}{-6}$.

      Now, the equation in standard symmetric form is:

      $\frac{x - 1/2}{2} = \frac{y - 1}{-3} = \frac{z - 0}{-6}$ …

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2. CBSE 2026Set 65/2/11 markMCQ
   Q.Assertion (A): A line can have direction cosines $< 1, 1, 1 >$. Reason (R): $\cos \theta = 1$ is possible for $\theta = 0$. (A) Both Assertion (A) and Reason (R) are true and the 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, but Reason (R) is false. (D) Assertion (A) is false, but Reason (R) is true.

   ›Reveal solutionSolution

   A line’s direction cosines must satisfy $l^2 + m^2 + n^2 = 1$. Since $1^2+1^2+1^2 = 3 \neq 1$, the triple $<1,1,1>$ cannot be direction cosines. So Assertion (A) is false. Reason (R) is true because $\cos 0 = 1$, but it does not explain (A). The correct option is (D).

   ---

   The core idea here is the definition of direction cosines. Direction cosines of a line are the cosines of the angles the line makes with the coordinate axes. If a line makes angles $\alpha, \beta, \gamma$ with the $x, y, z$ axes respectively, then its direction cosines are $l = \cos\alpha$, $m = \cos\beta$, $n = \cos\gamma$.

   A fundamental property — and the one that decides this question — is that these three numbers always satisfy $l^2 + m^2 + n^2 = 1$. Why? Because the direction vector of the line has components proportional to $l, m, n$, and its magnitude squared equals $l^2 + m^2 + n^2$ times some scale factor; but since $l, m, n$ are themselves the cosines, the vector $(\cos\alpha, \cos\beta, \cos\gamma)$ is a unit vector. So the sum of squares must be exactly 1.

   Now let’s examine the Assertion and Reason separately.

   1. Check Assertion (A): Can $<1, 1, 1>$ be direction cosines?

      Compute $1^2 + 1^2 + 1^2 = 3$. This is not equal to 1. Therefore $<1,1,1>$ violates the necessary condition. So the Assertion is false.

   2. Check Reason (R): Is $\cos\theta = 1$ possible?

      Yes, $\cos 0 = 1$. So the statement “$\cos\theta = 1$ is possible for $\theta = 0$” is true.

   3. Does Reason (R) explain Assertion (A)? …

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3. CBSE 2026Set V11 markMCQ
   Q.The position vector of the midpoint of the line joining the points $P(2, 3, 4)$ and $Q(4, 1, -2)$

   (a) $3\hat{i}+2\hat{j}+\hat{k}$

   (b) $3\hat{i}+2\hat{j}-\hat{k}$

   (c) $\hat{i}-\hat{j}-3\hat{k}$

   (d) $-\hat{i}+\hat{j}+3\hat{k}$

   ›Reveal solutionSolution

   Averaging the coordinates of $P$ and $Q$ gives $(3,2,1)$; answer (a).

   The position vector of the midpoint is the average of the two position vectors: …

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4. CBSE 2026Set CX1 mark
   Q.If a line makes $90^{\circ}$, $60^{\circ}$ and $30^{\circ}$ with $x$, $y$ and $z$-axes in the positive direction respectively, then find direction cosines.

   ›Reveal solutionSolution

   The direction cosines are just the cosines of the given angles: $\left(0,\tfrac12,\tfrac{\sqrt3}{2}\right)$.

   Concept: If a line makes angles $\alpha,\beta,\gamma$ with the $x,y,z$-axes, its direction cosines are $l=\cos\alpha,\ m=\cos\beta,\ n=\cos\gamma$.

   $l=\cos90^{\circ}=0,\qquad m=\cos60^{\circ}=\dfrac{1}{2},\qquad n=\cos30^{\circ}=\dfrac{\sqrt3}{2}.$

   …

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5. CBSE 2026Set A1 markMCQ
   Q.The direction ratios of a straight line are $2, 6, -3$. Then its direction cosines are

   (a) $\frac{1}{7}, \frac{2}{7}, \frac{3}{7}$

   (b) $\frac{2}{7}, \frac{-6}{7}, \frac{3}{7}$

   (c) $\frac{2}{7}, \frac{6}{7}, \frac{-3}{7}$

   (d) none of these

   ›Reveal solutionSolution

   Direction cosines $=$ direction ratios divided by their magnitude.

   Direction ratios are $2, 6, -3$. Their magnitude is

   $\sqrt{2^2 + 6^2 + (-3)^2} = \sqrt{4 + 36 + 9} = \sqrt{49} = 7.$ …

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6. CBSE 2026Set A1 markMCQ
   Q.If a line makes angles $\alpha$, $\beta$ and $\gamma$ with the positive directions of $x$, $y$ and $z$ axes respectively, then

   (a) $\cos^2\alpha + \cos^2\beta + \cos^2\gamma = 1$

   (b) $\sin^2\alpha + \sin^2\beta + \sin^2\gamma = 4$

   (c) $\cos^2\alpha + \cos^2\beta + \cos^2\gamma = 2$

   (d) $\sin^2\alpha + \sin^2\beta + \sin^2\gamma = 1$

   ›Reveal solutionSolution

   For direction cosines, $\cos^2\alpha + \cos^2\beta + \cos^2\gamma = 1$.

   If a line makes angles $\alpha,\beta,\gamma$ with the axes, then $l=\cos\alpha$, $m=\cos\beta$, $n=\cos\gamma$ are its direction cosines and satisfy $l^2+m^2+n^2 = 1$, i.e.

   $\cos^2\alpha + \cos^2\beta + \cos^2\gamma = 1.$ …

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7. CBSE 2026Set ANNUAL1 markMCQ
   Q.If $2\vec{a}+3\vec{b}-5\vec{c}=\vec{0}$, then write the ratio in which $\vec{c}$ divides $\overline{AB}$, where the position vectors of $A$ and $B$ are respectively $\vec{a}$ and $\vec{b}$.

   (a) 3 : 2 internally

   (b) 3 : 2 externally

   (c) 2 : 3 internally

   (d) 2 : 3 externally

   ›Reveal solutionSolution

   Rearranging $2\vec a+3\vec b-5\vec c=\vec 0$ shows $\vec c$ is the point dividing $\overline{AB}$ internally in the ratio $3:2$.

   We are given $2\vec a+3\vec b-5\vec c=\vec 0$, i.e.

   $5\vec c=2\vec a+3\vec b \quad\Rightarrow\quad \vec c=\frac{2\vec a+3\vec b}{5}=\frac{3\vec b+2\vec a}{3+2}$

   Section formula: if a point $C$ divides $\overline{AB}$ internally in the ratio $m:n$ (i.e. $AC:CB=m:n$), its position vector is

   $\vec c=\frac{n\vec a+m\vec b}{m+n}$

   …

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8. CBSE 2026Set ANNUAL1 markMCQ
   Q.If a line makes angles of $30^{\circ}$ and $45^{\circ}$ with $X$-axis and $Y$-axis respectively, then what is the angle made by it with $Z$-axis?

   (a) $45^{\circ}$

   (b) $60^{\circ}$

   (c) $120^{\circ}$

   (d) Cannot be determined

   ›Reveal solutionSolution

   Applying the direction-cosine identity to the given angles gives a negative value for $\cos^2\gamma$, which is impossible — so the required angle cannot exist / be determined from the given data.

   For a line making angles $\alpha,\beta,\gamma$ with the $X$-, $Y$-, $Z$-axes respectively, the direction cosines $l=\cos\alpha,\ m=\cos\beta,\ n=\cos\gamma$ must satisfy

   $l^2+m^2+n^2=1$

   Given $\alpha=30^{\circ}$, $\beta=45^{\circ}$:

   $\cos^2 30^{\circ}=\left(\frac{\sqrt3}{2}\right)^2=\frac34,\qquad \cos^2 45^{\circ}=\left(\frac{1}{\sqrt2}\right)^2=\frac12$

   So

   $n^2=\cos^2\gamma=1-\frac34-\frac12=1-\frac54=-\frac14$

   …

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9. CBSE 2026Set ANNUAL1 mark
   Q.Find the direction cosines of the line passing through the two points $(-2, 4, -5)$ and $(1, 2, 3)$.

   ›Reveal solutionSolution

   Find direction ratios from the two points, then divide by their magnitude to get direction cosines.

   Direction ratios: $(1-(-2),\,2-4,\,3-(-5)) = (3,-2,8)$.

   Magnitude $=\sqrt{3^2+(-2)^2+8^2}=\sqrt{9+4+64}=\sqrt{77}$.

   …

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10. CBSE 2026Set ANNUAL1 markMCQ
    Q.If a line makes angles $\alpha, \beta, \gamma$ with coordinate axes then $\sin^2\alpha + \sin^2\beta + \sin^2\gamma =$

    (a) 2

    (b) 1

    (c) -2

    (d) 0

    ›Reveal solutionSolution

    The direction cosines of a line satisfy $\cos^2\alpha+\cos^2\beta+\cos^2\gamma=1$; convert to sines using $\sin^2\theta=1-\cos^2\theta$.

    Since $\alpha,\beta,\gamma$ are the angles a line makes with the coordinate axes, its direction cosines satisfy:

    $\cos^2\alpha+\cos^2\beta+\cos^2\gamma = 1$ …

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11. CBSE 2026Set ANNUAL1 mark
    Q.Direction cosines of $y$-axis is ...........

    ›Reveal solutionSolution

    The $y$-axis makes angles $90°, 0°, 90°$ with the $x$, $y$, $z$ axes respectively.

    The direction cosines of a line are $(\cos\alpha,\cos\beta,\cos\gamma)$, the cosines of the angles it makes with the positive $x$, $y$, $z$ axes.

    …

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12. CBSE 2026Set ANNUAL1 markMCQ
    Q.Assertion (A): If a vector makes equal angle with co-ordinate axis then the direction cosines of the vector are $\pm\left(\dfrac{1}{\sqrt3}, \dfrac{1}{\sqrt3}, \dfrac{1}{\sqrt3}\right)$. Reason (R): A vector makes $\alpha$, $\beta$, $\gamma$ angle with positive direction on $x$, $y$ and $z$ axis respectively, then their direction cosines are $\cos\alpha, \cos\beta, \cos\gamma$.

    (a) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).

    (b) Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of Assertion (A).

    (c) Assertion (A) is true, but Reason (R) is false.

    (d) Assertion (A) is false, but Reason (R) is true.

    ›Reveal solutionSolution

    Equal angles with the axes plus the identity $\cos^2\alpha+\cos^2\beta+\cos^2\gamma=1$ together give the direction cosines in Assertion, using the definition in Reason.

    Reason (R): If a vector makes angles $\alpha,\beta,\gamma$ with the positive $x$, $y$, $z$ axes, its direction cosines are $\cos\alpha,\cos\beta,\cos\gamma$ — this is the standard definition, so R is true.

    …

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