List of top Mathematics Questions on Vector Algebra asked in KEAM

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

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

If \( \vec{a}, \vec{b} \) and \( \vec{c} \) are three non-zero vectors such that each one of them being perpendicular to the sum of the other two vectors, then the value of \( |\vec{a} + \vec{b} + \vec{c}|^2 \) is:

Let \( \vec{u}, \vec{v} \) and \( \vec{w} \) be vectors such that \( \vec{u} + \vec{v} + \vec{w} = \vec{0} \). If \( |\vec{u}| = 3, |\vec{v}| = 4 \) and \( |\vec{w}| = 5 \) then \( \vec{u} \cdot \vec{v} + \vec{v} \cdot \vec{w} + \vec{w} \cdot \vec{u} = \)

If \( \lambda(3\hat{i} + 2\hat{j} - 6\hat{k}) \) is a unit vector, then the values of \( \lambda \) are:

If \( \vec{a} \cdot \vec{b} = 0 \) and \( \vec{a} + \vec{b} \) makes an angle of \( 60^\circ \) with \( \vec{a} \), then:

If \( \lambda(3\hat{i} + 2\hat{j} - 6\hat{k}) \) is a unit vector, then the values of \( \lambda \) are:

If \( \vec{a} \cdot \vec{b} = 0 \) and \( \vec{a} + \vec{b} \) makes an angle of \( 60^\circ \) with \( \vec{a} \), then:

If \( \vec{a}, \vec{b} \) and \( \vec{c} \) are three non-zero vectors such that each one of them being perpendicular to the sum of the other two vectors, then the value of \( |\vec{a} + \vec{b} + \vec{c}|^2 \) is:

Let \( \vec{u}, \vec{v} \) and \( \vec{w} \) be vectors such that \( \vec{u} + \vec{v} + \vec{w} = \vec{0} \). If \( |\vec{u}| = 3, |\vec{v}| = 4 \) and \( |\vec{w}| = 5 \) then \( \vec{u} \cdot \vec{v} + \vec{v} \cdot \vec{w} + \vec{w} \cdot \vec{u} = \)

Let \( \vec{u}, \vec{v} \) and \( \vec{w} \) be vectors such that \( \vec{u} + \vec{v} + \vec{w} = \vec{0} \). If \( |\vec{u}| = 3, |\vec{v}| = 4 \) and \( |\vec{w}| = 5 \) then \( \vec{u} \cdot \vec{v} + \vec{v} \cdot \vec{w} + \vec{w} \cdot \vec{u} = \)

If \( \lambda(3\hat{i} + 2\hat{j} - 6\hat{k}) \) is a unit vector, then the values of \( \lambda \) are:

If \( \vec{a} \cdot \vec{b} = 0 \) and \( \vec{a} + \vec{b} \) makes an angle of \( 60^\circ \) with \( \vec{a} \), then:

If \( \vec{a}, \vec{b} \) and \( \vec{c} \) are three non-zero vectors such that each one of them being perpendicular to the sum of the other two vectors, then the value of \( |\vec{a} + \vec{b} + \vec{c}|^2 \) is: