If $|\vec{x}| = |\vec{y}| = |\vec{x} + \vec{y}| = 1$,then $|\vec{x} - \vec{y}| = $ . . . . . . .

$A$ unit vector in the $xy$-plane that makes an angle of $45^{\circ}$ with the vector $(i + j)$ and an angle of $60^{\circ}$ with the vector $(3i - 4j)$ is:

$AB=a$ and $AC=b$ are the sides of $\triangle ABC$. $P$ is a point on $AB$ and $Q$ is a point on $BC$ such that $\frac{AP}{PB}=\frac{1}{2}$ and $\frac{BQ}{QC}=\frac{1}{2}$. If the point of intersection of $AQ$ and $CP$ is $D$ and the area of $\triangle BCD$ is $7$ square units,then the area of the $\triangle ABC$ (in the same square units) is

If the coordinates of the points $P$ and $Q$ are $(1, -2, 1)$ and $(2, 3, 4)$ respectively,and $O$ is the origin $(0, 0, 0)$,then which of the following is true?

If the position vectors of the points $A$ and $B$ are $2\,i + 3\,j - k$ and $-2\,i + 3\,j + 4\,k$,then the line $AB$ is parallel to

Find the distance of the point $P(-\hat{i} + 2\hat{j} + 6\hat{k})$ from the line passing through the point $A(2\hat{i} + 3\hat{j} - 4\hat{k})$ and parallel to the vector $\vec{b} = 6\hat{i} + 3\hat{j} - 4\hat{k}$.