What are the values of $x$ for which the angle between the vectors $2x^2\hat{i} + 3x\hat{j} + \hat{k}$ and $\hat{i} - 2\hat{j} + x^2\hat{k}$ is obtuse? [Application]
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Vectors - ISC Class 12 Mathematics Questions with Answers, Page 2
55 past-paper questions on Vectors from ISC Class 12 Mathematics papers (2026-2017), newest first, in full. Questions 21-40 are on this page, 20 to a page. Tap "Show answer" under a question to see its answer.
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Find a vector of magnitude 9 units and perpendicular to the vectors $\vec{a} = 4\hat{i} - \hat{j} + \hat{k}$ and $\vec{b} = -2\hat{i} + \hat{j} - 2\hat{k}$. [Application]
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Let $\vec{\alpha} = 3\hat{i} + \hat{j}$ and $\vec{\beta} = 2\hat{i} - \hat{j} + 3\hat{k}$. If $\vec{\beta} = \vec{\beta}_1 - \vec{\beta}_2$ where $\vec{\beta}_1$ is parallel to $\vec{\alpha}$ and $\vec{\beta}_2$ is perpendicular to $\vec{\alpha}$.
(a)
Find $\vec{\beta}_1$.
(b)
Find $\vec{\beta}_2$.
(c)
Hence, find $\vec{\beta}_1 \times \vec{\beta}_2$.
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The value of $\hat{i} \cdot (\hat{k} \times \hat{j}) + \hat{j} \cdot (\hat{i} \times \hat{k}) + \hat{k} \cdot (\hat{i} \times \hat{j})$ is:
- (a)$-3$
- (b)$-2$
- (c)$-1$
- (d)$0$
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If $\vec{p} \times \vec{q} = \vec{0}$ and $\vec{p} \cdot \vec{q} = 0$, then what conclusion can we draw?
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Assertion: $(\vec{a} + \vec{b}) \cdot (\vec{b} - \vec{a}) = 2 (a^2 + b^2)$
Reason: Dot product of any two vectors is commutative.
- (a)Both Assertion and Reason are true and Reason is the correct explanation for Assertion.
- (b)Both Assertion and Reason are true but Reason is not the correct explanation for Assertion.
- (c)Assertion is true and Reason is false.
- (d)Assertion is false and Reason is true.
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If the angle between $\vec{a} = -3\hat{i} + x\hat{j} + \hat{k}$ and $\vec{b} = x\hat{i} + 2x\hat{j} + \hat{k}$ is acute and the angle between $\vec{b}$ and the x-axis lies between $\frac{\pi}{2}$ and $\pi$. Find the range of values for $x$.
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Derive a condition for which given, $\vec{b} = \vec{c}$, we can say $\vec{a} \times \vec{b} = \vec{c} \times \vec{a}$?
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In subparts (i) and (ii) choose the correct options and in subparts (iii) to (v), answer the questions as instructed.
(i)[1.0]
If $\vec{a} = 3\hat{i} - 2\hat{j} + \hat{k}$ and $\vec{b} = 2\hat{i} - 4\hat{j} - 3\hat{k}$ then the value of $|\vec{a} - 2\vec{b}|$ will be:
- (a)$\sqrt{85}$
- (b)$\sqrt{86}$
- (c)$\sqrt{87}$
- (d)$\sqrt{88}$
(ii)[1.0]
If a line makes an angle $\alpha$, $\beta$ and $\gamma$ with positive direction of the coordinate axes, then the value of $\sin^2\alpha + \sin^2\beta + \sin^2\gamma$ will be:
- (a)1
- (b)3
- (c)-2
- (d)2
(iii)[1.0]
In the figure given below, if the coordinates of the point $P$ are $(a, b, c)$, then what are the perpendicular distances of $P$ from $XY$, $YZ$ and $ZX$ planes respectively?
(iv)[1.0]
If $\vec{a} = 2\hat{i} + \hat{j} + 2\hat{k}$ and $\vec{b} = 5\hat{i} - 3\hat{j} + \hat{k}$, find the projection of $\vec{b}$ on $\vec{a}$.
(v)[1.0]
Find a vector of magnitude 20 units parallel to the vector $2\hat{i} + 5\hat{j} + 4\hat{k}$.
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If $\vec{a} \times \vec{b} = \vec{a} \times \vec{c}$ where $\vec{a}$, $\vec{b}$ and $\vec{c}$ are non-zero vectors, then prove that either $\vec{b} = \vec{c}$ or $\vec{a}$ and $(\vec{b} - \vec{c})$ are parallel.
Given: $\vec{a} \times \vec{b} = \vec{a} \times \vec{c}$
To show: $\vec{b} = \vec{c}$ or $\vec{a}$ and $(\vec{b} - \vec{c})$ are parallel
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If $\vec{a}$ and $\vec{b}$ are two non-zero vectors such that $|\vec{a} \times \vec{b}| = \vec{a} \cdot \vec{b}$, find the angle between $\vec{a}$ and $\vec{b}$.
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If the two vectors $3\hat{i} + a\hat{j} + \hat{k}$ and $2\hat{i} - \hat{j} + 8\hat{k}$ are perpendicular to each other, then find the value of $a$.
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If $A(1, 2, -3)$ and $B(-1, -2, 1)$ are the end points of a vector $\vec{AB}$ then find the unit vector in the direction of $\vec{AB}$.
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For any two non-zero vectors $\vec{a}$ and $\vec{b}$, if $|\vec{a} \cdot \vec{b}| = |\vec{a} \times \vec{b}|$, then find the angle between them.
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Find the area of the parallelogram whose diagonals are $\hat{i} - 3\hat{j} + \hat{k}$ and $\hat{i} + \hat{j} + \hat{k}$.
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If $|\vec{a}| = 3$, $|\vec{b}| = \frac{\sqrt{2}}{3}$ and $\vec{a} \times \vec{b}$ is a unit vector then the angle between $\vec{a}$ and $\vec{b}$ will be:
- (a)$\frac{\pi}{6}$
- (b)$\frac{\pi}{4}$
- (c)$\frac{\pi}{3}$
- (d)$\frac{\pi}{2}$
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If $\hat{a}$ is unit vector and $(2\vec{x} - 3\hat{a}) \cdot (2\vec{x} + 3\hat{a}) = 91$, find the value of $|\vec{x}|$.
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Find the area of the triangle whose adjacent sides are $\hat{i} + 4\hat{j} - \hat{k}$ and $\hat{i} + \hat{j} + 2\hat{k}$.
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The given figure shows an air plant holder which is in the shape of a tetrahedron. Let $A(1, 1, 1)$, $B(2, 1, 3)$, $C(3, 2, 2)$ & $D(3, 3, 4)$ are the vertices of air plant holder.
Based on the above information answer the following questions.
(i)[2.0]
The vector of $\vec{AB}$ is:
- (a)$-\hat{i} - 2\hat{k}$
- (b)$2\hat{i} + \hat{k}$
- (c)$\hat{i} + 2\hat{k}$
- (d)$-2\hat{i} - \hat{k}$
(ii)[2.0]
The vector of $\vec{AC}$ is:
- (a)$2\hat{i} - \hat{j} - \hat{k}$
- (b)$2\hat{i} + \hat{j} + \hat{k}$
- (c)$-2\hat{i} - \hat{j} + \hat{k}$
- (d)$\hat{i} + 2\hat{j} + \hat{k}$
(iii)[2.0]
Area of $\Delta ABC$ is:
- (a)$\frac{\sqrt{11}}{2}\text{ Sq. units}$
- (b)$\frac{\sqrt{14}}{2}\text{ Sq. units}$
- (c)$\frac{\sqrt{13}}{2}\text{ Sq. units}$
- (d)$\frac{\sqrt{17}}{2}\text{ Sq. units}$
(iv)[2.0]
The unit vector along the $\vec{AB}$ is:
- (a)$\frac{-2\hat{i} - \hat{k}}{\sqrt{5}}$
- (b)$\frac{-\hat{i} - 2\hat{k}}{\sqrt{5}}$
- (c)$\frac{2\hat{i} + \hat{k}}{\sqrt{5}}$
- (d)$\frac{\hat{i} + 2\hat{k}}{\sqrt{5}}$
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What will be the value of m if the vector $2\hat{i} + m\hat{j} + \hat{k}$ is perpendicular to $2\hat{i} - \hat{j} + 3\hat{k}$?
- (a)7
- (b)0
- (c)1
- (d)-1
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