Consider two non-zero vectors and in three-dimensional space.

MATHEMATICS · Multi-part · CBSE · Mathematics · Vector Algebra · Vector Algebra

Problem

Consider two non-zero vectors

\mathbf{a}

and

\mathbf{b}

in three-dimensional space. (i) State the relation between

\mathbf{a}\cdot\mathbf{b}

|\mathbf{a}|

|\mathbf{b}|

and the angle

\theta

between the vectors. (ii) Prove that if

\mathbf{a}\cdot\mathbf{b}=|\mathbf{a}|\,|\mathbf{b}|

, then

\mathbf{a}

and

\mathbf{b}

are parallel and point in the same direction. (iii) Prove that if

\mathbf{a}\cdot\mathbf{b}=-|\mathbf{a}|\,|\mathbf{b}|

, then

\mathbf{a}

and

\mathbf{b}

are parallel but point in opposite directions. (iv) Give one concrete numerical example in

\mathbb R^{3}

to illustrate each of the results obtained in parts (ii) and (iii).