The position vector of the point which divides the line segment joining the points with position vectors $2a - 3b$ and $3a - 2b$ internally in the ratio $2 : 3$ is:

Which of the following is not a unit vector for all values of $\theta$?

$A$ vector of magnitude $\sqrt{2}$ units along the internal bisector of the angle between the vectors $\vec{a} = 2 \hat{i} - 2 \hat{j} + \hat{k}$ and $\vec{b} = \hat{i} + 2 \hat{j} + 2 \hat{k}$ is

If a vector $a$ of magnitude $50$ is collinear with the vector $b = 6i - 8j - \frac{15}{2}k$, and makes an acute angle with the positive direction of the $z$-axis, then the vector $a$ is equal to:

In the given figure,which of the vectors are equal?

Let a vector $\alpha \hat{i}+\beta \hat{j}$ be obtained by rotating the vector $\sqrt{3} \hat{i}+\hat{j}$ by an angle $45^{\circ}$ about the origin in the counterclockwise direction in the first quadrant. Then the area of the triangle having vertices $(\alpha, \beta), (0, \beta)$ and $(0,0)$ is equal to