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Vectors - ISC Class 12 Mathematics Questions with Answers, Page 3

55 past-paper questions on Vectors from ISC Class 12 Mathematics papers (2026-2017), newest first, in full. Questions 41-55 are on this page, 20 to a page. Tap "Show answer" under a question to see its answer.

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2021 · 5 marks · Short answerOpen: [5×1] In sub-parts (i) and (ii) choose the correct options and in sub-parts…
[5×1] In sub-parts (i) and (ii) choose the correct options and in sub-parts (iii) to (v), answer the questions as instructed.
(i)
If the vectors $a\hat{i} + 3\hat{j} - 2\hat{k}$ and $3\hat{i} - 4\hat{j} + b\hat{k}$ are collinear, then $(a, b) =$
  • (a)$\left(\frac{9}{4}, \frac{8}{3}\right)$
  • (b)$\left(-\frac{9}{4}, \frac{8}{3}\right)$
  • (c)$\left(\frac{9}{4}, -\frac{8}{3}\right)$
  • (d)$\left(-\frac{9}{4}, -\frac{8}{3}\right)$
(ii)
The intercepts made by the plane $3x - 2y + 4z = 12$ on the coordinate axes are:
  • (a)$6, -4, 3$
  • (b)$4, -6, 3$
  • (c)$2, -3, 4$
  • (d)$\frac{1}{4}, -\frac{1}{6}, \frac{1}{3}$
(iii)
Find the angle between the vectors $\vec{a} = 6\hat{i} + 2\hat{j} + 3\hat{k}$, $\vec{b} = 2\hat{i} - 9\hat{j} + 6\hat{k}$.
(iv)
Find the unit vector parallel to the vector: $3\hat{i} + 6\hat{j} - 2\hat{k}$
(v)
Find the equation of the plane passing through $(-2, 1, 3)$ and perpendicular to the line having direction ratios $\langle 3, 1, 5 \rangle$.

No answer yet.

2020 · 4 marks · DerivationOpen: Prove that .
Prove that $\vec{a} \cdot [(\vec{b} + \vec{c}) \times (\vec{a} + 3\vec{b} + 4\vec{c})] = [\vec{a} \quad \vec{b} \quad \vec{c}]$.

Given: $\vec{a} \cdot [(\vec{b} + \vec{c}) \times (\vec{a} + 3\vec{b} + 4\vec{c})]$

To show: $[\vec{a} \quad \vec{b} \quad \vec{c}]$

No answer yet.

2020 · 6 marks · Short answerOpen: [3×2] Write a vector of magnitude of 18 units in the direction of the vector …
[3×2]
(a)[2.0]
Write a vector of magnitude of 18 units in the direction of the vector $\hat{i} - 2\hat{j} - 2\hat{k}$.
(b)[2.0]
Find the angle between the two lines: $\frac{x + 1}{2} = \frac{y - 2}{5} = \frac{z + 3}{4} \quad \text{and} \quad \frac{x - 1}{5} = \frac{y + 2}{2} = \frac{z - 1}{-5}$
(c)[2.0]
Find the equation of the plane passing through the point $(2, -3, 1)$ and perpendicular to the line joining the points $(4, 5, 0)$ and $(1, -2, 4)$.

No answer yet.

2019 · 4 marks · DerivationOpen: If , , prove that and are perpendicular.
If $\vec{a} = -\hat{i} - 2\hat{j} + 3\hat{k}$, $\vec{b} = 2\hat{i} + 3\hat{j} - 5\hat{k}$, prove that $\vec{a}$ and $\vec{a} \times \vec{b}$ are perpendicular.

No answer yet.

2018 · 4 marks · DerivationOpen: Show that:
Show that: $(\vec{a} \times \vec{b})^2 = \begin{vmatrix} \vec{a} \cdot \vec{a} & \vec{a} \cdot \vec{b} \\ \vec{a} \cdot \vec{b} & \vec{b} \cdot \vec{b} \end{vmatrix}$

Given: $(\vec{a} \times \vec{b})^2$

To show: $\begin{vmatrix} \vec{a} \cdot \vec{a} & \vec{a} \cdot \vec{b} \\ \vec{a} \cdot \vec{b} & \vec{b} \cdot \vec{b} \end{vmatrix}$

No answer yet.

2018 · 4 marks · DerivationOpen: Show that:
Show that: $\vec{a} \cdot (\vec{b} + \vec{c}) \times (\vec{a} + 2\vec{b} + 3\vec{c}) = [\vec{a} \; \vec{b} \; \vec{c}]$

Given: $\vec{a} \cdot (\vec{b} + \vec{c}) \times (\vec{a} + 2\vec{b} + 3\vec{c})$

To show: $[\vec{a} \; \vec{b} \; \vec{c}]$

No answer yet.

2018 · 4 marks · DerivationOpen: Show that the four points and with position vectors , , and respectively, are…
Show that the four points $A, B, C$ and $D$ with position vectors $4\hat{\imath} + 5\hat{\jmath} + \hat{k}$, $-\hat{\jmath} - \hat{k}$, $3\hat{\imath} + 9\hat{\jmath} + 4\hat{k}$ and $4(-\hat{\imath} + \hat{\jmath} + \hat{k})$ respectively, are coplanar.

Given: Four points $A, B, C, D$ with position vectors $4\hat{\imath} + 5\hat{\jmath} + \hat{k}$, $-\hat{\jmath} - \hat{k}$, $3\hat{\imath} + 9\hat{\jmath} + 4\hat{k}$ and $4(-\hat{\imath} + \hat{\jmath} + \hat{k})$

To show: Points $A, B, C, D$ are coplanar

No answer yet.

2018 · 6 marks · Short answerOpen: [3×2] Find if the scalar projection of on is units. The Cartesian equation of a…
[3×2]
(a)[2.0]
Find $\lambda$ if the scalar projection of $\vec{a} = \lambda\hat{\imath} + \hat{\jmath} + 4\hat{k}$ on $\vec{b} = 2\hat{\imath} + 6\hat{\jmath} + 3\hat{k}$ is $4$ units.
(b)[2.0]
The Cartesian equation of a line is: $2x - 3 = 3y + 1 = 5 - 6z$. Find the vector equation of a line passing through $(7, -5, 0)$ and parallel to the given line.
(c)[2.0]
Find the equation of the plane through the intersection of the planes $\vec{r} \cdot (\hat{\imath} + 3\hat{\jmath} - \hat{k}) = 9$ and $\vec{r} \cdot (2\hat{\imath} - \hat{\jmath} + \hat{k}) = 3$ and passing through the origin.

No answer yet.

2018 · 4 marks · DerivationOpen: If are three non-collinear points with position vectors , respectively, then…
If $A, B, C$ are three non-collinear points with position vectors $\vec{a}, \vec{b}, \vec{c}$, respectively, then show that the length of the perpendicular from $C$ on $AB$ is $\frac{|(\vec{a} \times \vec{b}) + (\vec{b} \times \vec{c}) + (\vec{c} \times \vec{a})|}{|\vec{b} - \vec{a}|}$.

Given: $A, B, C$ are three non-collinear points with position vectors $\vec{a}, \vec{b}, \vec{c}$, respectively

To show: Length of the perpendicular from $C$ on $AB$ is $\frac{|(\vec{a} \times \vec{b}) + (\vec{b} \times \vec{c}) + (\vec{c} \times \vec{a})|}{|\vec{b} - \vec{a}|}$

No answer yet.

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