$A$ unit vector in the $xy$-plane which is perpendicular to $4i - 3j + k$ is

If $a \cdot b = 0$ and $a + b$ makes an angle $60^{\circ}$ with $a$,then

If the projection of the vector $\hat{i}+2 \hat{j}+\hat{k}$ on the sum of the two vectors $2 \hat{i}+4 \hat{j}-5 \hat{k}$ and $-\lambda \hat{i}+2 \hat{j}+3 \hat{k}$ is $1,$ then $\lambda$ is equal to ..... .

If $\vec{\alpha} = 3\hat{i} - \hat{k}$, $|\vec{\beta}| = \sqrt{5}$, and $\vec{\alpha} \cdot \vec{\beta} = 3$, then the area of the parallelogram for which $\vec{\alpha}$ and $\vec{\beta}$ are adjacent sides is:

If the position vectors of the vertices of a triangle are $2 \hat{i}-\hat{j}+\hat{k}$,$\hat{i}-3 \hat{j}-5 \hat{k}$,and $3 \hat{i}-4 \hat{j}-4 \hat{k}$,then the triangle is

Let $ABCD$ be a parallelogram such that $\vec{AB} = \vec{q}$ and $\vec{AD} = \vec{p}$,and $\angle BAD$ is an acute angle. If $\vec{r}$ is a vector that coincides with the altitude from vertex $B$ to the side $AD$,find $\vec{r}$.