If the vectors $\vec{AB} = -3\hat{i} + 4\hat{k}$ and $\vec{AC} = 5\hat{i} - 2\hat{j} + 4\hat{k}$ are the sides of a $\triangle ABC$, then the length of the median through $A$ is

The vector having magnitude of $2 \sqrt{29}$ units in the direction of vector $\vec{a} = 4 \hat{i} + 3 \hat{j} - 2 \hat{k}$ is . . . . . . .

In a quadrilateral $ABCD$, the point $P$ divides $DC$ in the ratio $1:3$ internally and $Q$ is the mid-point of $AC$. If $\vec{AB} + \vec{AD} + \vec{BC} - 2\vec{DC} = \lambda \vec{PQ}$, then the value of $\lambda$ is

Let $ABC$ be a triangle whose circumcentre is at $P$. If the position vectors of $A, B, C$ and $P$ are $\vec{a}, \vec{b}, \vec{c}$ and $\frac{\vec{a} + \vec{b} + \vec{c}}{4}$ respectively,then the position vector of the orthocentre of this triangle is:

If the vector $\vec{a} = \hat{i} + \hat{j} + \hat{k}$ makes angles $\alpha, \beta, \gamma$ with vectors $\hat{i}, \hat{j}, \hat{k}$ respectively,then:

The position vectors of $P$ and $Q$ are $5i + 4j + ak$ and $-i + 2j - 2k$ respectively. If the distance between them is $7$,then the value of $a$ will be: