Vector Algebra

If a vector $\vec{v}=3 \hat{i}$ is rotated in the $x-z$ plane by an angle $\theta$ with respect to $x$－axis in the clockwise direction，then for an observer at $+y$ axis the vector will be

Three vectors $\vec{a}, \vec{b}$ and $\vec{c}$ are such that $|\vec{a}|=1,|\vec{b}|=2$ and $|\vec{c}|=4$ along with $\vec{a}+\vec{b}+\vec{c}=0$ ．Then，the value of $4 \vec{a} \cdot \vec{b}+3 \vec{b} \cdot \vec{c}+3 \vec{c} \cdot \vec{a}$ wil...

Six vectors $\vec{a}, \vec{b}, \vec{c}, \vec{d}, \vec{e}$ and $\vec{f}$ have the magnitudes and directions indicated in the figure. Which of the following statements is true?

Let \(\theta\) be the angle between two vectors \(\vec{A}\) and \(\vec{B}\). If \(\hat{a}_{\perp}\) is the unit vector perpendicular to \(\vec{A}\), then the direction of $$
\overrightarrow{\mathrm{B}}-\mathrm{B} \sin \theta \hat{\mathrm{a}...

Consider the vectors \(A = \hat i + \hat j - \hat k\)
,\(B = 2\hat i - \hat j + \hat k\) and \(C = {1 \over {\sqrt 5 }}\left( {\hat i - 2\hat j + 2\hat k} \right)\)
. What is the value of C. (A\(\times\) B) ?

In a triangle ABC, the sides AB and AC are represented by the vectors \(3\widehat i + \widehat j + \widehat k\) and \(\widehat i + 2\widehat j + \widehat k\), respectively. Calculate the angle \(\angle ABC\).

Three vectors \(\overrightarrow A\) = a\(\widehat i\) + \(\widehat j\) + \(\widehat k\); \(\overrightarrow B\) = \(\widehat i\) + b\(\widehat j\) + \(\widehat k\) and \(\overrightarrow C\) = \(\widehat i\) + \(\widehat j\) + c$$\widehat ...

The vectors \(\overrightarrow A\) and \(\overrightarrow B\) are such that | \(\overrightarrow A\) + \(\overrightarrow B\) | = | \(\overrightarrow A\) \(-\) \(\overrightarrow B\) |. The angle between the two vectors will be

The angle subtended by the vector \(A = 4\widehat i + 3\widehat j + 12\widehat k\) with the x-axis is