Consider two non-zero vectors and in three–dimensional space so that the angle…

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

Problem

Consider two non-zero vectors

\mathbf{u}

and

\mathbf{v}

in three–dimensional space so that the angle between them is denoted by

\theta

. (i) State the scalar product of

\mathbf{u}

and

\mathbf{v}

in terms of

|\mathbf{u}|,\,|\mathbf{v}|

and

\theta

. (ii) Explain how the sign of

\mathbf{u}\!\cdot\!\mathbf{v}

determines whether the angle

\theta

is acute, right or obtuse. (iii) A student claims that if

\mathbf{u}\!\cdot\!\mathbf{v}=-5

, then

\theta

lies between

90^{\circ}

and

180^{\circ}

. Justify this claim without calculating

\theta

explicitly.