If $a = 3i - 2j + k$,$b = 2i - 4j - 3k$ and $c = -i + 2j + 2k$,then $a + b + c$ is

Let $OA = a, OB = b$ be two non-collinear vectors, $OP = x_1 a + y_1 b, OQ = x_2 a + y_2 b$ and $A^{\prime}O = OA, B^{\prime}O = OB$. If $x_1 = -\frac{3}{4}, x_2 = \frac{1}{3}, y_1 = \frac{7}{4}, y_2 = \frac{5}{3}$, then

Let $\alpha, \beta, \gamma$ be distinct real numbers. The points with position vectors $\alpha \hat{i} + \beta \hat{j} + \gamma \hat{k}$,$\beta \hat{i} + \gamma \hat{j} + \alpha \hat{k}$,and $\gamma \hat{i} + \alpha \hat{j} + \beta \hat{k}$:

The direction cosine of the vector $\vec{a} = 3\hat{i} + 4\hat{j} + 5\hat{k}$ in the direction of the positive $x$-axis is:

Answer the following as true or false.
$\vec{a}$ and $-\vec{a}$ are collinear.

Let $b = 3j + 4k$,$a = i + j$ and let $b_1$ and $b_2$ be component vectors of $b$ parallel and perpendicular to $a$ respectively. If $b_1 = \frac{3}{2}i + \frac{3}{2}j$,then $b_2 = $