Q.Find the position vector of a point $A$ in space such that $\vec{OA}$ is inclined at $60^\circ$ to $OX$ and at $45^\circ$ to $OY$ and $|\vec{OA}| = 10$ units.

Showing the 12 most recent of 84 on this concept.

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:

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

   > [!WARNING]
   > A common mistake is to directly take $(4, 3, 6)$ or $(4, -3, -6)$ as direction ratios. This is incorrect because the numerators were not in the standard $(x-x_1)$, $(y-y_1)$, $(z-z_1)$ form. Always ensure the coefficient of $x, y, z$ in the numerator is $+1$.
   

   2. Identify the direction ratios.

   From the standard form $\frac{x - 1/2}{2} = \frac{y - 1}{-3} = \frac{z - 0}{-6}$, the direction ratios $(a, b, c)$ are the denominators.

   So, $a = 2$, $b = -3$, $c = -6$.

   3. Calculate the magnitude of the direction vector.

      The magnitude is $\sqrt{a^2 + b^2 + c^2}$.

      Magnitude $= \sqrt{(2)^2 + (-3)^2 + (-6)^2}$

      Magnitude $= \sqrt{4 + 9 + 36}$

      Magnitude $= \sqrt{49}$

      Magnitude $= 7$.

   4. Calculate the direction cosines.

      The direction cosines $(l, m, n)$ are obtained by dividing each direction ratio by the magnitude.

      $l = \frac{a}{\text{Magnitude}} = \frac{2}{7}$

      $m = \frac{b}{\text{Magnitude}} = \frac{-3}{7}$

      $n = \frac{c}{\text{Magnitude}} = \frac{-6}{7}$

      So, the direction cosines are $\left(\frac{2}{7}, \frac{-3}{7}, \frac{-6}{7}\right)$.

      Comparing this with the given options, option (B) matches our result.

   ✓Final answer

   The direction cosines of the given line are $\boxed{\frac{2}{7}, \frac{-3}{7}, \frac{-6}{7}}$.

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)?

      The Reason talks about a single cosine being 1, but the Assertion claims a triple of three 1’s can be direction cosines. Even if $\cos\theta = 1$ is possible, that doesn’t make $<1,1,1>$ valid — because the sum-of-squares condition fails. So (R) is not the correct explanation of (A).

   Watch out

   A common mistake is to think that because each individual number 1 is a possible cosine (for angle $0^\circ$), the triple must be valid. But direction cosines are not independent — they must satisfy $l^2+m^2+n^2=1$. Three 1’s break that rule.

   Thus, Assertion (A) is false, Reason (R) is true, and (R) does not explain (A). That matches option (D).

   ✓Final answer

   The correct option is (D).

3. 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 $l=\cos\alpha,\ m=\cos\beta,\ n=\cos\gamma$.

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

   Check: $l^{2}+m^{2}+n^{2}=0+\dfrac14+\dfrac34=1$ ✓ (as required for direction cosines).

   ✓Final answer

   Direction cosines $=\left(0,\ \dfrac{1}{2},\ \dfrac{\sqrt3}{2}\right)$.

4. 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

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

   So the direction cosines are $\dfrac{2}{7}, \dfrac{6}{7}, \dfrac{-3}{7}$.

   ✓Final answer

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

5. 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.

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

   (Equivalently $\sin^2\alpha + \sin^2\beta + \sin^2\gamma = 2$, not $1$, so option (d) is wrong.)

   ✓Final answer

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

6. 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

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

   This is negative, which is impossible for any real $\cos^2\gamma\ge0$. Hence no real angle $\gamma$ satisfies the given combination of $30^{\circ}$ and $45^{\circ}$ with the other two axes — such a line does not exist, so the angle with the $Z$-axis cannot be determined.

   ✓Final answer

   The correct option is (d) Cannot be determined (the given pair of angles is inconsistent with $l^2+m^2+n^2=1$).

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

   Direction cosines $=\left(\dfrac{3}{\sqrt{77}},\dfrac{-2}{\sqrt{77}},\dfrac{8}{\sqrt{77}}\right)$.

   ✓Final answer

   Direction cosines $=\left(\dfrac{3}{\sqrt{77}},\dfrac{-2}{\sqrt{77}},\dfrac{8}{\sqrt{77}}\right)$.

8. 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:

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

   So $\sin^2\alpha+\sin^2\beta+\sin^2\gamma = (1-\cos^2\alpha)+(1-\cos^2\beta)+(1-\cos^2\gamma) = 3 - 1 = 2$.

   ✓Final answer

   (a) $2$.

9. 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.

   The $y$-axis makes $\alpha=90°$ with the $x$-axis, $\beta=0°$ with itself, and $\gamma=90°$ with the $z$-axis.

   So the direction cosines are $(\cos90°,\cos0°,\cos90°)=(0,1,0)$.

   ✓Final answer

   $(0, 1, 0)$

10. 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.

    Assertion (A): If the vector makes equal angles with all three axes, $\alpha=\beta=\gamma$, so its direction cosines are $l=m=n=\cos\alpha$. Using the identity $l^2+m^2+n^2=1$: $3\cos^2\alpha=1 \Rightarrow \cos\alpha=\pm\dfrac{1}{\sqrt3}$. So the direction cosines are $\pm\left(\dfrac{1}{\sqrt3},\dfrac{1}{\sqrt3},\dfrac{1}{\sqrt3}\right)$ — A is true.

    R (the definition of direction cosines) is exactly what is used to derive A, so R correctly explains A.

    ✓Final answer

    Option (a): both true, R is the correct explanation of A.

11. CBSE 2026Set ANNUAL1 markMCQ
    Q.Direction cosines of x-axis are

    (a) <0, 0, 0>

    (b) <1, 1, 1>

    (c) <0, 0, 1>

    (d) <1, 0, 0>

    ›Reveal solutionSolution

    The x-axis makes a $0°$ angle with itself and $90°$ with both the y- and z-axes.

    Direction cosines are $(\cos\alpha,\cos\beta,\cos\gamma)$ where $\alpha,\beta,\gamma$ are the angles the line makes with the x-, y-, z-axes respectively.

    For the x-axis itself: $\alpha=0°,\ \beta=90°,\ \gamma=90°$, so

    $(\cos0°,\cos90°,\cos90°) = (1,0,0).$

    (Check: $1^2+0^2+0^2=1$, satisfying $l^2+m^2+n^2=1$ as required for direction cosines.)

    ✓Final answer

    $\langle 1,0,0\rangle$. (Option d)

12. CBSE 2026Set ANNUAL1 markMCQ
    Q.Assertion (A): If a line has direction ratios −18, 12, −4 then its direction cosines are −9/11, 6/11, −2/11. Reason (R): If a line has direction ratios a, b, c then its direction cosines are a/√(a²+b²+c²), b/√(a²+b²+c²), c/√(a²+b²+c²).

    (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) Both Assertion (A) and Reason (R) are false.

    ›Reveal solutionSolution

    Dividing the given direction ratios by their magnitude reproduces exactly the direction cosines stated in the Assertion, confirming both statements and the explanation link.

    Reason (R) states the standard formula: if direction ratios are $a,b,c$, the direction cosines are $\dfrac{a}{\sqrt{a^2+b^2+c^2}},\dfrac{b}{\sqrt{a^2+b^2+c^2}},\dfrac{c}{\sqrt{a^2+b^2+c^2}}$ — this is the correct general formula, so R is true.

    Checking Assertion (A): ratios are $a=-18,\ b=12,\ c=-4$.

    $\sqrt{a^2+b^2+c^2} = \sqrt{324+144+16} = \sqrt{484} = 22.$

    Direction cosines $= \left(\dfrac{-18}{22},\dfrac{12}{22},\dfrac{-4}{22}\right) = \left(-\dfrac{9}{11},\dfrac{6}{11},-\dfrac{2}{11}\right).$

    This exactly matches what A claims, so A is true, and it was verified by directly applying the formula in R — so R is the correct explanation of A.

    ✓Final answer

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