Vector Algebra

Let $\mathbf{a}=\hat{\mathbf{i}}-\hat{\mathbf{k}}, \mathbf{b}=x \hat{\mathbf{i}}+\hat{\mathbf{j}}+(1-x) \hat{\mathbf{k}}$ and $\mathbf{c}=y \hat{\mathbf{i}}+x \hat{\mathbf{j}}+(1+x-y) \hat{\mathbf{k}}$. Then,…

If $\vec{a}=2 \hat{i}+\hat{j}+2 \hat{k}$, then the value of $|\hat{i} \times(\vec{a} \times \hat{i})|^2+|\hat{j} \times(\vec{a} \times \hat{j})|^2+|\hat{k} \times(\vec{a} \times \hat{k})|^2$ is equal to

Let ABC be a triangle and be $\vec{a}, \vec{b}, \vec{c}$ the position vectors of $\mathrm{A}, \mathrm{B}, \mathrm{C}$ respectively. Let D divide $B C$ in the ratio $3: 1$ internally and $E$ divide AD in the ratio $4: 1$…

A vector perpendicular to the plane containing the vectors $\hat{i}-2 \hat{j}-\hat{k}$ and $3 \hat{i}-2 \hat{j}-\hat{k}$ is inclined to the vector $\hat{i}+\hat{j}+\hat{k}$ at an angle

If $\vec{a}=\hat{i}+\hat{j}+\hat{k}, \vec{a} \cdot \vec{b}=1$ and $\vec{a} \times \vec{b}=\vec{j}-\vec{k}$, then $\vec{b}$ is

$\hat{\mathbf{u}}$ and $\hat{\mathbf{v}}$ are two non-collinear unit vectors such that $\left|\frac{\hat{\mathbf{u}}+\hat{\mathbf{v}}}{\mathbf{2}}+\hat{\mathbf{u}} \times \hat{\mathbf{v}}\right|=1$. Then the value of…

If $\vec{a}$ is a vector of magnitude 50 , collinear with the vector $\vec{b}=6 \hat{i}-8 \hat{j}-\frac{15}{2} \hat{k}$ and makes an acute angle with the positive direction of $Z$ - axis, then $\vec{a}$ is equal to

If $\vec{r}$ and $\vec{s}$ arenon-zero constant vectors and the scalar $b$ is chosen such that $|\vec{r}+b \vec{s}|$ is minimum, then the value of $|b \vec{s}|^{2}+|\vec{r}+b \vec{s}|^{2}$ is equal to

If $\vec{a}=\vec{i}+2\vec{j}+3\vec{k}, \vec{b}=-\vec{i}+2\vec{j}+\vec{k}, \vec{c}=3\vec{i}+\vec{j}$ and $\vec{a}+p\vec{b}$ is normal to $\vec{c}$, then $p$ is equal to

If $\vec{a}, \vec{b}, \vec{c}$ are non-coplaner vectors such that $\vec{b}×\vec{c}=\vec{a}; \vec{c}×\vec{a}=\vec{b}; \vec{a}×\vec{b}=\vec{c}$, then which of the following is not TRUE?

If $h$ is the altitude of a parallelopiped determined by the vectors $\vec{a}, \vec{b}, \vec{c}$ and the base is taken to be the parallelogram determined by $\vec{a}$ and $\vec{b}$ where…

Let $\vec{a}$ and $\vec{b}$ are two non-collinear unit vectors. If $\vec{u}=\vec{a}-(\vec{a}·\vec{b})\vec{b}$ and $\vec{v}=\vec{a}×\vec{b}$, then $|\vec{v}|$ is equal to