List of top Mathematics Questions on Vector Algebra asked in KEAM

Let the three vectors $\vec{a}, \vec{b}$, and $\vec{c}$ be pairwise non-collinear vectors. If $\vec{a}+2\vec{b}$ is collinear with $\vec{c}$ and if $\vec{b}+2\vec{c}$ is collinear with $\vec{a}$, then $\vec{a}+2\vec{b}+5\vec{c}$ is equal to
Let $\vec{OP}=2\hat{i}-2\hat{j}-\hat{k}$ and $\vec{OQ}=2\hat{i}+\hat{j}+2\hat{k}$. If the point $R$ lies on $\vec{PQ}$ and $\vec{OR}$ bisects the angle $\angle POQ$, then $2\vec{OR}$ is} \textit{Note: The initial vector has been mathematically corrected from the exam's typo ($2\hat{i}-2\hat{j}-2\hat{k}$) to standard format ($2\hat{i}-2\hat{j}-\hat{k}$) to permit a valid solution.
If $\vec{a}\times(\hat{i}-\hat{j}+\hat{k})=(\hat{i}-\hat{j}+\hat{k})\times\vec{b}$ and $|\vec{a}+\vec{b}|=3\sqrt{3}$, then the possible values of $(\vec{a}+\vec{b})\cdot(3\hat{i}+2\hat{j}+\hat{k})$ are

If $ | \vec{a} | = 3 $, $ | \vec{b} | = 2 $, then find $ (3\vec{a} - 2\vec{b}) \cdot (3\vec{a} + 2\vec{b}) $.

Given that \( \mathbf{a} \times (2\hat{i} + 3\hat{j} + 4\hat{k}) = (2\hat{i} + 3\hat{j} + 4\hat{k}) \times \mathbf{b} \), \( |\mathbf{a} + \mathbf{b}| = \sqrt{29} \), \( \mathbf{a} \cdot \mathbf{b} = ? \)
Given that \( \mathbf{a} \times (2\hat{i} + 3\hat{j} + 4\hat{k}) = (2\hat{i} + 3\hat{j} + 4\hat{k}) \times \mathbf{b} \), \( |\mathbf{a} + \mathbf{b}| = \sqrt{29} \), \( \mathbf{a} \cdot \mathbf{b} = ? \)
Given that \( |\mathbf{a} + \mathbf{b}| = \frac{\sqrt{14}}{2} \), where \( \mathbf{a} \) and \( \mathbf{b} \) are unit vectors, find the value of \( |\mathbf{a} + \mathbf{b}|^2 - |\mathbf{a} - \mathbf{b}|^2 \).
Given that \( \vec{a} \parallel \vec{b} \), \( \vec{a} \cdot \vec{b} = \frac{49}{2} \), and \( |\vec{a}| = 7 \), find \( |\vec{b}| \).
If ABCD is a rectangle, $ AB = 5i + 4j - 3k $ and $ AD = 3i + 2j - k $, then find $ BD $.
Let \( \alpha, \beta, \gamma \) be the direction cosines of a vector \( \vec{a} = x \hat{i} + y \hat{j} + z \hat{k} \), where \( z<0 \). If \( \alpha = \frac{-4}{\sqrt{105}} \) and \( \beta = \frac{\sqrt{5}}{\sqrt{21}} \), then \( \gamma \) is equal to
Let \( A(0,3,-3) \), \( B(1,1,1) \) and \( C(2,0,3) \) be three points in space. Then the projection of \( \overrightarrow{AB} \) on \( \overrightarrow{AC} \) is equal to
If \( \vec{a} = 5\hat{i} - 7\hat{j} + 9\hat{k} \) and \( \vec{b} = -5\hat{i} + 7\hat{j} - 9\hat{k} \), then \( \vec{a} \cdot (\vec{a} \times \vec{b}) + (\vec{a} + \vec{b}) \cdot \hat{b} \) is equal to
The angle between the vectors \( \vec{a} \) and \( \vec{b} \) is \( \frac{\pi}{3} \). If \( | \vec{a} \cdot \vec{b} |^2 = 15 \), then \( | \vec{a} \times \vec{b} |^2 \) is equal to

Let, \(a=i-j+2k\). Then the vector in the direction of a with magnitude \(5\) units is ?

If \(|\bar u|=3,|\bar v|=2\ and\ |\bar u\times \bar v|=3\), then the angle between \(\bar u\ and\ \bar v\) is equal to
If \(\bar a\ and\ \bar b\) are position vectors of the points (a, 3, 0) and (1, 0, 0) respectively and if the angle between the vectors \(\bar a\ and\ \bar b\ is\ \frac{\pi}{4}\), then the value of a is equal to
If \(\bar{a}=\hat{i}-3\hat{j}+3\hat{k}\) and \(\bar{b}=2\hat{i}+\hat{j}-3\hat{k}\), then the value of \((\bar a\times\bar b)\cdot\bar b\) is equal to
If \(|\bar a|=10\ and\ |\bar b|=5\), then the value of \((\bar a+2\bar b)\cdot(\bar a-2\bar b)\) is equal to
If \(|\bar{a}|=\sqrt{14},|\bar{b}|=\sqrt{10},|\bar{a}-\bar{b}|=\sqrt{24}\ and\ \theta\) is angle between \(\bar{a}\ and\ \bar{b}\), then \(\cos\theta=\)
Let \(\bar{a}=\hat{i}+2\hat{j}-3\hat{k}\) and \(\bar{a}+\bar{b}=4\hat{i}-2\hat{j}+\lambda\hat{k}\). If \(\bar{a}\cdot\bar{b}=4\), then the value of \(\lambda\) is equal to
The angle between the line \(\overrightarrow{r}=\hat{i}+2\hat{j}+t(3\hat{i}+2\hat{j}-\hat{k})\) and the plane 2x-3y-z=1 is
If \(|\overrightarrow{u}|=5,|\overrightarrow{v}|=4\) and the angle between \(\overrightarrow{u}\) and \(\overrightarrow{v}\) is \(\frac{\pi}{6}\), then \(|\overrightarrow{u}\times\overrightarrow{v}|\) is equal to
The values of a so that the vectors \(a\hat{i}+(a-1)\hat{j}+3\hat{k}\) and \((a+2)\hat{i}+a\hat{j}-2\hat{k}\) are perpendicular, are
If \(\overrightarrow{a}=\hat{i}-3\hat{j}+\alpha\hat{k}, \overrightarrow{b}=\hat{i}-2\hat{j}+4\hat{k}\ and\ \overrightarrow{a}\times\overrightarrow{b}=-2\hat{i}+\hat{j}+\beta\hat{k}\), then the value of \(\beta\) is equal to
If \(|\overrightarrow{a}|=2,\overrightarrow{b}=2\hat{i}-\hat{j}-3\hat{k}\) and the angle between \(\overrightarrow{a}\) and \(\overrightarrow{b}\) is \(\frac{\pi}{4}\), then \(\overrightarrow{a}\cdot\overrightarrow{b}\) is equal to