List of top Mathematics Questions on Product of Two Vectors asked in KEAM

If the angle between the vectors $\vec{a}=x\hat{i}+3\hat{j}+\hat{k}$ and $\vec{b}=x\hat{i}-x\hat{j}+2\hat{k}$ is acute, then $x$ lies is

If $|\vec{a}| = \sqrt{26}, \; |\vec{b}| = \sqrt{3}$ and $\vec{a} \times \vec{b} = 5\hat{i} + \hat{j} - 4\hat{k}$, then $\vec{a} \cdot \vec{b} =$:

Let $\vec{a} = 2\hat{i} - 2\hat{j} + 4\hat{k}$, $\vec{b} = -5\hat{i} - \hat{j} + 8\hat{k}$ and $\vec{c} = 3\hat{i} + \hat{j} - \lambda \hat{k}$. If $\vec{a} + \vec{b} + \vec{c}$ and $\vec{a} - \vec{b} + \vec{c}$ are perpendicular, then the values of $\lambda$ are:

If $|\vec{a}| = 8$, $|\vec{b}| = 5$ and $|\vec{a} - \vec{b}| = 7$, then the angle between $\vec{a}$ and $\vec{b}$ is equal to

If $|\vec{a}| = 3$, $|\vec{b}| = 2$, then $(2\vec{a} + 3\vec{b})\cdot(2\vec{a} - 3\vec{b})$ is equal to

Let $\vec{a} = \alpha\,\hat{i} - 3\hat{j} - 2\hat{k}$ and $\vec{c} = \hat{i} - 2\hat{j} + 2\hat{k}$. If the projection of $\vec{a}$ on $\vec{c}$ is 2, then the value of $\alpha$ is

If \( |\vec{a}| = 12 \) and the projection of \( \vec{a} \) on \( \vec{b} \) is \( 6\sqrt{3} \), then the angle between \( \vec{a} \) and \( \vec{b} \) is

Let \( \vec{a} = 6\hat{i} + 2\hat{j} + 3\hat{k} \). If \( \vec{b} \) is parallel to \( \vec{a} \) and \( \vec{a} \cdot \vec{b} = \frac{49}{2} \), then \( |\vec{b}| = \)

If \( |\vec{a} + \vec{b}| = \frac{\sqrt{14}}{2} \) where \( \vec{a} \) and \( \vec{b} \) are unit vectors, then the value of \( |\vec{a} + \vec{b}|^2 - |\vec{a} - \vec{b}|^2 \) is equal to

Let \( \vec{a} + \vec{b} = \lambda \hat{i} + 16\hat{j} - 18\hat{k} \) and \( \vec{a} - \vec{b} = 2\hat{i} + 8\hat{j} + \lambda \hat{k} \). If \( \vec{a} + \vec{b} \) is perpendicular to \( \vec{a} - \vec{b} \), then \( |\vec{a}| = \)

A unit vector $\vec{b}$ is coplanar with $\hat{i} + \hat{j} + 2\hat{k}$ and $\hat{i} + 2\hat{j} + \hat{k}$ and is perpendicular to $\hat{i} + \hat{j} + \hat{k}$. Then $\vec{b} \cdot \hat{i}$ equals

The area of the triangle whose vertices are $A(1,-1,2)$, $B(2,1,-1)$ and $C(3,-1,2)$ is

A particle is displaced from the point \( (2,1,-1) \) to \( (4,3,-4) \) by the force \( 2i + 4j - 5k \). Then the work done by the force is

The value of \( m \) if the vectors \( 4i - 3j + 5k \) and \( mi - 4j + k \) are perpendicular, is

If \( |\vec{a}|=3, |\vec{b}|=1, |\vec{c}|=4 \) and \( \vec{a}+\vec{b}+\vec{c}=0 \), then the value of \( \vec{a}\cdot\vec{b} + \vec{b}\cdot\vec{c} + \vec{c}\cdot\vec{a} \) is

If the vectors \( \vec{a} = \hat{i} - \hat{j} + 2\hat{k}, \vec{b} = 2\hat{i} + 4\hat{j} + \hat{k} \) and \( \vec{c} = 2\lambda \hat{i} + 9\hat{j} + \mu \hat{k} \) are mutually orthogonal, then \( \lambda + \mu \) is equal to

Let \( \vec{a} = \hat{i}+\hat{j}+\hat{k}, \vec{b} = \hat{i}+3\hat{j}+5\hat{k}, \vec{c} = 7\hat{i}+9\hat{j}+11\hat{k} \). Then the area of parallelogram with diagonals \( \vec{a}+\vec{b} \) and \( \vec{b}+\vec{c} \) is

If \( |\vec{a}|=3, |\vec{b}|=1, |\vec{c}|=4 \) and \( \vec{a}+\vec{b}+\vec{c}=0 \), then the value of \( \vec{a}\cdot\vec{b} + \vec{b}\cdot\vec{c} + \vec{c}\cdot\vec{a} \) is

If \( |\vec{a}-\vec{b}| = |\vec{a}| = |\vec{b}| = 1 \), then the angle between \( \vec{a} \) and \( \vec{b} \) is

If the vectors \( \vec{a} = \hat{i} - \hat{j} + 2\hat{k}, \vec{b} = 2\hat{i} + 4\hat{j} + \hat{k} \) and \( \vec{c} = 2\lambda \hat{i} + 9\hat{j} + \mu \hat{k} \) are mutually orthogonal, then \( \lambda + \mu \) is equal to

If the vectors \( 4\hat{i} + 11\hat{j} + m\hat{k},\ 7\hat{i} + 2\hat{j} + 6\hat{k} \) and \( \hat{i} + 5\hat{j} + 4\hat{k} \) are coplanar, then \( m \) is equal to

Let \( \vec{a} = \hat{i}+\hat{j}+\hat{k}, \vec{b} = \hat{i}+3\hat{j}+5\hat{k}, \vec{c} = 7\hat{i}+9\hat{j}+11\hat{k} \). Then the area of parallelogram with diagonals \( \vec{a}+\vec{b} \) and \( \vec{b}+\vec{c} \) is

If \( \vec{a} \) and \( \vec{b}=3\hat{i}+6\hat{j}+6\hat{k} \) are collinear and \( \vec{a}\cdot\vec{b}=27 \), then \( \vec{a} \) is

If \( |\vec{a}|=13, |\vec{b}|=5 \) and \( \vec{a}\cdot\vec{b}=30 \), then \( |\vec{a}\times\vec{b}| \) is