Cross Product (Vector Product) PYQs

Find the vector product $(\hat{i} + 2\hat{j} + 3\hat{k}) \times (2\hat{i} - \hat{j} + \hat{k})$. 2 Marks

Find the area of the parallelogram whose adjacent sides are represented by the vectors $\hat{i} + \hat{j}$ and $\hat{i} - \hat{j}$. 2 Marks

Find the area of the triangle formed by the vectors $\vec{a} = \hat{i} + \hat{j} + \hat{k}$ and $\vec{b} = 2\hat{i} - \hat{j} + \hat{k}$. 2 Marks

Find a unit vector perpendicular to both the vectors $\hat{i} + \hat{j}$ and $\hat{i} - \hat{j}$. 3 Marks

Find the area of the triangle whose vertices are $(1,0,0), (0,1,0), (0,0,1)$. 4 Marks

Show that $(\vec{a} + \vec{b}) \times \vec{c} = \vec{a} \times \vec{c} + \vec{b} \times \vec{c}$. 3 Marks

Using vectors, find the area of the triangle whose vertices are $(1,2,3), (2,3,1), (3,1,2)$. 4 Marks

Using vector product, find the area of the parallelogram formed by vectors $\hat{i} + 2\hat{j} + \hat{k}$ and $2\hat{i} - \hat{j} + \hat{k}$. 2 Marks

Find $(\hat{i} + 2\hat{j} + 3\hat{k}) \times (2\hat{i} + \hat{j} - \hat{k})$. 2 Marks

Find $(2\hat{i} - \hat{j} + \hat{k}) \times (\hat{i} + \hat{j} - \hat{k})$. 2 Marks

Find the area of the parallelogram whose adjacent sides are $\hat{i} + \hat{j} + \hat{k}$ and $2\hat{i} - \hat{j} + \hat{k}$. 2 Marks

Find a unit vector perpendicular to both $\hat{i} + \hat{j} + \hat{k}$ and $2\hat{i} - \hat{j} + \hat{k}$. 3 Marks

If |a| = 8, |b| = 3 and |$\vec{a}$ × b| = 12, then value of |$\vec{a}·b$| is ______. 1 Mark

The unit vector perpendicular to the vectors $\hat{i}$−$\hat{j}$ and $\hat{i}+\hat{j}$ is ______. 1 Mark

Let $\vec{a}$ be any vector such that |a| = a. The value of |$\vec{a}×\hat{i}$|²+|$\vec{a}×\hat{j}$|²+|$\vec{a}×\hat{k}$|² is ______. 1 Mark

Find $\vec{a}$ vector of magnitude 4 units perpendicular to each of the vectors $2\hat{i}$−$\hat{j}+\hat{k}$ and $\hat{i}+\hat{j}$−$\hat{k}$ and hence verify your answer. 3 Marks

Given $\vec{a}$ = $2\hat{i}$−$\hat{j}+\hat{k}$, $\vec{b}$ = $3\hat{i}$−$\hat{k}$ and $\vec{c}$ = $2\hat{i}+\hat{j}$−$2\hat{k}$. Find $\vec{a}$ vector $\vec{d}$ which is perpendicular to both $\vec{a}$ and $\vec{b}$ and $\vec{c}·\vec{d}$ = 3. 3 Marks

Find λ and μ, if ($\hat{i}+3\hat{j}$+$9\hat{k}$)×($3\hat{i}$−λ$\hat{j}+$μ$\hat{k}$) = 0. 1 Mark

Write the number of vectors of unit length perpendicular to both the vectors $\vec{a}=2\hat{i}$+$\hat{j}+2\hat{k}$ and $\vec{b}=\hat{j}+\hat{k}$. 1 Mark

If |a|=2, |b|=7 and $\vec{a}×\vec{b}=3\hat{i}$+$2\hat{j}+6\hat{k}$, find the angle between $\vec{a}$ and b. 1 Mark

Find |$\vec{a}×b$|, if $\vec{a}=2\hat{i}$+$\hat{j}+5\hat{k}$ and $\vec{b}=3\hat{i}$+$5\hat{j}$–$2\hat{k}$. 1 Mark

If θ is the angle between two vectors $\hat{i}$–$2\hat{j}+3\hat{k}$ and $3\hat{i}$–$2\hat{j}+\hat{k}$, find sin θ. 1 Mark

Let $\vec{a}=4\hat{i}$+$5\hat{j}$–$\hat{k}$, $\vec{b}=\hat{i}$–$4\hat{j}+5\hat{k}$ and $\vec{c}=3\hat{i}$+$\hat{j}$–$\hat{k}$. Find $\vec{a}$ vector $\vec{d}$ which is perpendicular to both $\vec{c}$ and $\vec{b}$ and $\vec{d}·\vec{a}=21$. 1 Mark

Using vectors, find the area of the ΔABC, whose vertices are A(1,2,5), B(2,−1,4) and C(4,5,−1). 1 Mark

Show that the points A, B, C with position vectors $2\hat{i}$–$\hat{j}+\hat{k}$, $\hat{i}$–$5\hat{j}$–$5\hat{k}$ and $5\hat{i}$–$4\hat{j}$–$4\hat{k}$ respectively, are the vertices of $\vec{a}$ $right-angled$ triangle. Hence find the area of the triangle. 1 Mark

If $\vec{a}×\vec{b}=\vec{c}×\vec{d}$ and $\vec{a}×\vec{c}=\vec{b}×d$, then show that a−$\vec{d}$ is parallel to b−c, where a≠$\vec{d}$ and b≠c. 1 Mark