Motion in plane practice · Mixed difficulty

The area of the parallelogram represented by the vectors $\vec A\, = \,2\hat i\, + \,3\hat j$ and $\vec B\, = \,\hat i\, + \,4\hat j$ is

The direction cosines of a vector $\vec A$ are $\cos \alpha = \frac{4}{{5\sqrt 2 }},\cos \beta = \frac{1}{{\sqrt 2 }}$ and $\cos r = \frac{3}{{5\sqrt 2 }}$. Then the vector $\vec A$ is

A cricket ball thrown across a field is at heights $h_1$ and $h_2$ from the point of projection at times $t_1$ and $t_2$ respectively after the throw. The ball is caught by a fielder at the same height as that of projection. The time of flight of the ball in this journey is

Let $\mathbf{A}=\hat{\mathbf{i}} A \cos \theta+\hat{\mathbf{j}} A \sin \theta$ be any vector. Another vector $\mathbf{B}$ which is perpendicular to $\mathbf{A}$ can be expressed as

The component of a vector $\overrightarrow{{P}}=3 \hat{{i}}+8 \hat{{j}}$ along the direction $(\hat{{i}}+2 \hat{{j}})$ is

The speed of a projectile $v$ reduces by $50 \%$ on reaching maximum height. The range on the horizonta plane for that projectile is

The vector $\hat{\mathbf{i}}+\mathrm{x} \hat{\mathbf{j}}+3 \hat{\mathrm{k}}$ is rotated through an angle $\theta$ and doubled in magnitude, then it becomes $4 \hat{i}+(4 x-2) \hat{j}+2 \hat{k}$. The possible positive value of $x$ is

If $\mathbf{a} \times \mathbf{b}=\mathbf{b} \times \mathbf{c} \neq 0$, then the correct statement is

Which of the following is not essential for the three vectors to produce zero resultant?