Vector Algebra

Let $O$ be the origin and $OA\to=2i^+2j^+k^,OB\to=i^-2j^+2k^$ and $OC\to=\frac{1}{2}(OB\to-\lambda OA\to)$ for some $\lambda>0.$ If $|OB\to\times OC\to|=\frac{9}{2},$ then which of the following statements is (are) TRUE?

Let $a$ and $b$ be positive real numbers. Suppose $PQ\to=ai^+bj^$ and $PS\to=ai^-bj^$ are adjacent sides of a parallelogram $PQRS.$ Let $u\to$ and $v\to$ be the projection vectors of $w\to=i^+j^$ along $PQ\to$ and…

Paragraph: Let $O$ be the origin, and $\overrightarrow{O X}, \overrightarrow{O Y}, \overrightarrow{O Z}$ be three unit vectors in the directions of the sides $\overrightarrow{Q R}, \overrightarrow{R P}$,…

Let $O$ be the origin and let $PQR$ be an arbitrary triangle. The point$S$ is such that $OP\to.OQ\to+OR\to.OS\to=OR\to.OP\to+OQ\to.OS\to=OQ\to.OR\to+OP\to.OS\to$ then triangle $PQR$ has $S$ as its

Let $\text{PR}\to=3i⁡^+j⁡^-2k^\text{and}\text{SQ}\to=i⁡^-3j⁡^-4k^$ determine diagonals of a parallelogram PQRS and $\text{PT}\to=i⁡^+2j⁡^+3k^$ be another vector. Then the volume of the parallelepiped determined by the…

$$ \text { Match the statements of Column I with values of Column II. } $$

Two adjacent sides of a parallelogram $A B C D$ are given by $\overrightarrow{\mathbf{A B}}=2 \hat{\mathbf{i}}+10 \hat{\mathbf{j}}+11 \hat{\mathbf{k}}$ and $\overrightarrow{\mathbf{A D}}=-\hat{\mathbf{i}}+2…

Let $P, Q, R$ and $S$ be the points on the plane with position vectors $-2 \hat{\mathbf{i}}-\hat{\mathbf{j}}, 4 \hat{\mathbf{i}}, 3 \hat{\mathbf{i}}+3 \hat{\mathbf{j}}$ an $\mathrm{d}-3 \hat{\mathbf{i}}+2…

Match the statements/expressions given in Column I with the values given in Column II.

If $\mathbf{a}, \mathbf{b}, \mathbf{c}$ and $\mathbf{d}$ are the unit vectors such that $(\mathbf{a} \times \mathbf{b}) \cdot(\mathbf{c} \times \mathbf{d})=1$ and $\mathbf{a} \cdot \mathbf{c}=\frac{1}{2}$, then

The edges of a parallelopiped are of unit length and are parallel to non-coplanar unit vectors $\hat{\mathbf{a}}, \hat{\mathbf{b}}, \hat{\mathbf{c}}$ such that $\hat{\mathbf{a}} \cdot \hat{\mathbf{b}}=\hat{\mathbf{b}}…

Let the vectors $\mathbf{P Q}, \mathbf{Q R}, \mathbf{R S}, \mathbf{S T}, \mathbf{T U}$ and UP represent the sides of a regular hexagon. Statement I PQ $\times(\mathbf{R S}+\mathbf{S T}) \neq \mathbf{0}$ Statement II…