Vector Algebra

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…

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?

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…

Let $\vec{a}, \vec{b}$ and $\vec{c}$ be three mutually perpendicular vectors each of unit magnitud. If $\vec{A}=\vec{a}+\vec{b}+\vec{c}, \vec{B}=\vec{a}-\vec{b}+\vec{c}$ and…

Let $\vec{a}$ and $\vec{b}$ be two unit vectors such that $|\vec{a}-\vec{b}| \lt 2$. If $2 \theta$ is the angle between $\vec{a}$ and $\vec{b}$, then which one of the following is correct?

Two adjacent sides of a parallelogram are $2 \hat{i}-4 \hat{j}+5 \hat{k}$ and $\hat{i}-2 \hat{j}-3 \hat{k}$. What is the magnitude of dot product of vectors which represent its diagonals?

Let $\vec{a}, \vec{b}$ and $\vec{c}$ be three vectors such that $\vec{a}, \vec{b}$ and $\vec{c}$ are coplanar. Which of the following is /are correct? $(\vec{a} \times \vec{b}) \times \vec{c}$ is coplanar with $\vec{a}$…

If are the position vectors of the vertices $A, B, C$ respectively of a triangle $A B C$ and $G$ is the centroid of the triangle, then what is equal to?

Let $\theta$ be the angle between two unit vectors $\vec{a}$ and $\vec{b}$. If $\vec{a}+2 \vec{b}$ is perpendicular to $5 \vec{a}-4 \vec{b}$, then what is $\cos \theta+$ $\cos 2 \theta$ equal to?

For the following two (02) items: Let $A(1, -1, 0)$, $B(-2, 1, 8)$ and $C(-1, 2, 7)$ are three consecutive vertices of a parallelogram $ABCD$. If angle $BCD$ is $\theta$, then what is $\cos^2\theta$ equal to?

Consider the following statements : Dot product over vector addition is distributive Cross product over vector addition is distributive Cross product of vectors is associative Which of the above statements is/are…

What is the scalar projection of $\vec{a}=\hat{i}-2 \hat{j}+\hat{k}$ on $\vec{b}=4 \hat{i}-4 \hat{j}+7 \hat{k}$ ?