Vector Algebra

$P Q R S$ is a parallelogram. If $\overrightarrow{P R}=\vec{a}$ and $\overrightarrow{Q S}=\vec{b}$, then what is $\overrightarrow{P Q}$ equal to?

If $\hat{a}$ is a unit vector in the $x y$-plane making an angle $30^{\circ}$ with the positive $x$-axis, then what is $\hat{a}$ equal to ?

What is the area of the triangle whose vertices are $(0,0,0),(1,2,3)$ and $(-3,-2,1)$ ?

$A B C D$ is a parallelogram and $P$ is the point of intersection of the diagonals. If $O$ is the origin, then…

Let $\overrightarrow{\mathrm{p}}$ and $\overrightarrow{\mathrm{q}}$ be the position vectors of the points P and Q respectively with respect to origin $O$. The points $R$ and $S$ divide PQ internally and externally…

If in a right-angled triangle ABC , hypotenuse $\mathrm{AC}=\mathrm{p}$, then what is $\overrightarrow{A B} \cdot \overrightarrow{A C}+\overrightarrow{B C} \cdot \overrightarrow{B A}+\overrightarrow{C A} \cdot…

A vector $\vec{r}=a \hat{i}+b \hat{j}$ is equally inclined to both $x$ and $y$ axes. If the magnitude of the vector is 2 units, then what are the values of $a$ and $b$ respectively?

What is the area of the parallelogram having diagonals $3 \hat{\mathrm{i}}+\hat{\mathrm{j}}-2 \hat{\mathrm{k}}$ and $\hat{\mathrm{i}}-3 \hat{\mathrm{j}}+4 \hat{\mathrm{k}}$ ?

Let $A B C D$ be a parallelogram whose diagonals intersect at P and let O be the origin. What is $\overrightarrow{\mathrm{OA}}+\overrightarrow{\mathrm{OB}}+\overrightarrow{\mathrm{OC}}+\overrightarrow{\mathrm{OD}}$…

In a right-angled triangle ABC , if the hypotenuse $\mathrm{AB}=\mathrm{p}$, then what is $\overrightarrow{\mathrm{AB}} \cdot \overrightarrow{\mathrm{AC}}+\overrightarrow{\mathrm{BC}} \cdot…

ABCD is a parallelogram. If $\overrightarrow{\mathrm{AB}}=\vec{a}, \overrightarrow{\mathrm{BC}}=\vec{b}$, then what is $\overrightarrow{\mathrm{BD}}$ equal to?

Let ABCDEF be a regular hexagon. If $\overrightarrow{\mathrm{AD}}=m \overrightarrow{\mathrm{BC}}$ and $\overrightarrow{\mathrm{CF}}=n \overrightarrow{\mathrm{AB}}$, then what is $m n$ equal to?