In subparts (i) and (ii) choose the correct options and in subparts (iii) to (v), answer the questions as instructed.
(i)[1.0]
If $\vec{a} = \hat{i} + 2\hat{j} + 3\hat{k}$ and $\vec{b} = -\hat{i} + 2\hat{j} + \hat{k}$ and $\vec{c} = 3\hat{i} + \hat{j}$, find $t$ such that $\vec{a} + t\vec{b}$ is perpendicular to $\vec{c}$ is
(ii)[1.0]
The planes $2x - y + 4z = 5$ and $5x - 2\cdot 5y + 10z = 6$ are
- (a)parallel
- (b)intersect on y axis
- (c)perpendicular
- (d)pass through $(0, 0, \frac{5}{4})$
(iii)[1.0]
Find a vector of magnitude of $10$ units and parallel to the vector $2\hat{i} + 3\hat{j} - \hat{k}$.
(iv)[1.0]
Find the position vector of a point R which divides the line joining the two-points $P$ and $Q$ with position vectors $2\hat{i} + \hat{j}$ and $\hat{i} - 2\hat{j}$ respectively in the ratio of $2:1$ externally.
(v)[1.0]
Find the equation of the plane with intercept $3$ on the $y\text{--axis}$ and parallel to $xz\text{--plane}$.
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