List of top Mathematics Questions on Vector basics asked in KEAM

Let $\theta$ be the angle between the unit vectors $\hat{a}$ and $\hat{b}$. If $|\hat{a} - \hat{b}| = \frac{\sqrt{3}}{2}$, then the value of $\cos \theta$ is

Let \(|\vec{a}| = 6\) and \(|\vec{b}| = 10\). If \(\vec{a}\) and \(\vec{b}\) make angles \(25^\circ\) and \(85^\circ\), respectively, with the x-axis, then the value of \(|\vec{a} + \vec{b}|\) is equal to

If $2\hat{i} - \hat{j} + \hat{k} = s(3\hat{i} - 4\hat{j} - 4\hat{k}) + t(\hat{i} - 3\hat{j} - 5\hat{k})$, where $s$ and $t$ are scalars, then $3s + 5t$ is equal to:

Let $\vec{OA}=2\hat{i}+3\hat{j}-5\hat{k}$, $\vec{OB}=3\hat{i}+\hat{j}-2\hat{k}$, $\vec{OC}=6\hat{i}-5\hat{j}+7\hat{k}$ be position vectors of A, B and C. Then ________.

Let \(\vec{a}\) and \(\vec{b}\) be two unit vectors, and \(\theta\) be the angle between them. If \(\vec{a}-\vec{b}\) is a unit vector, then \(\theta\) is equal to

The unit vector that bisects the angle between two vectors \( 2\hat{i}+\hat{j}+2\hat{k} \) and \( \hat{i}+2\hat{j}-2\hat{k} \) is

Let $\vec{a}, \vec{b}, \vec{c}$ be any three vectors and $m, n$ be scalars. Which one of the following is not true?

The points with position vectors \(60\hat{i}+3\hat{j}, 40\hat{i}-8\hat{j}, a\hat{i}-52\hat{j}\) are collinear if

Suppose $\alpha \hat{i} + \alpha \hat{j} + \gamma \hat{k}$, $\hat{i} + \hat{k}$ and $\gamma \hat{i} + \gamma \hat{j} + \beta \hat{k}$ are coplanar where $\alpha, \beta, \gamma$ are positive constants. Then the product $\alpha\beta$ is

Let the position vectors of points \(A, B, C\) be \( \vec{a}, \vec{b}, \vec{c} \) respectively. Let \(Q\) be the centroid. Then \( \overrightarrow{QA} + \overrightarrow{QB} + \overrightarrow{QC} = \)

If \( \vec{a} = \hat{i} + \hat{j} + \hat{k}, \vec{b} = 4\hat{i} + 3\hat{j} + 4\hat{k} \) and \( \vec{c} = \hat{i} + \alpha \hat{j} + \beta \hat{k} \) are coplanar and \( |\vec{c}| = \sqrt{3} \), then

The angle between a normal to the plane \( 2x - y + 2z - 1 = 0 \) and the \( z \)-axis is:

Equation of the plane through the mid-point of the line segment joining the points P(4, 5, -10) and Q(-1, 2, 1) and perpendicular to PQ is:

Let \( \vec{a} = \hat{i} - 2\hat{j} + 3\hat{k} \). If \( \vec{b} \) is a vector such that \( \vec{a} \cdot \vec{b} = |\vec{b}|^2 \) and \( |\vec{a} - \vec{b}| = \sqrt{7} \), then \( |\vec{b}| = \)

The angle between a normal to the plane \( 2x - y + 2z - 1 = 0 \) and the \( z \)-axis is:

Equation of the plane through the mid-point of the line segment joining the points P(4, 5, -10) and Q(-1, 2, 1) and perpendicular to PQ is:

Let \( \vec{a} = \hat{i} - 2\hat{j} + 3\hat{k} \). If \( \vec{b} \) is a vector such that \( \vec{a} \cdot \vec{b} = |\vec{b}|^2 \) and \( |\vec{a} - \vec{b}| = \sqrt{7} \), then \( |\vec{b}| = \)

The angle between a normal to the plane \( 2x - y + 2z - 1 = 0 \) and the \( z \)-axis is:

Equation of the plane through the mid-point of the line segment joining the points P(4, 5, -10) and Q(-1, 2, 1) and perpendicular to PQ is:

Let \( \vec{a} = \hat{i} - 2\hat{j} + 3\hat{k} \). If \( \vec{b} \) is a vector such that \( \vec{a} \cdot \vec{b} = |\vec{b}|^2 \) and \( |\vec{a} - \vec{b}| = \sqrt{7} \), then \( |\vec{b}| = \)