Classical Mechanics (Kibble & Berkshire) · Problem 1.1

Velocity, acceleration and scalar angular velocity — T.W.B. Kibble, F.H. Berkshire

Physics — Mechanics · vector-algebra-scalar-product-cross-product

Problem

In terms of the standard basis set

\{\boldsymbol{i}, \boldsymbol{j}, \boldsymbol{k}\}

\boldsymbol{a} = 2\boldsymbol{i} - \boldsymbol{j} - 2\boldsymbol{k}

\boldsymbol{b} = 3\boldsymbol{i} - 4\boldsymbol{k}

and

\boldsymbol{c} = \boldsymbol{i} - 5\boldsymbol{j} + 3\boldsymbol{k}

. (i) Find

3\boldsymbol{a} + 2\boldsymbol{b} - 4\boldsymbol{c}

and

|\boldsymbol{a} - \boldsymbol{b}|^2

. (ii) Find

|\boldsymbol{a}|

|\boldsymbol{b}|

and

\boldsymbol{a} \cdot \boldsymbol{b}

. Deduce the angle between

\boldsymbol{a}

and

\boldsymbol{b}

. (iii) Find the component of

\boldsymbol{c}

in the direction of

\boldsymbol{a}

and in the direction of

\boldsymbol{b}

. (iv) Find

\boldsymbol{a}\times\boldsymbol{b}

\boldsymbol{b}\times\boldsymbol{c}

and

(\boldsymbol{a}\times\boldsymbol{b})\times(\boldsymbol{b}\times\boldsymbol{c})

. (v) Find

\boldsymbol{a} \cdot (\boldsymbol{b}\times\boldsymbol{c})

and

(\boldsymbol{a}\times\boldsymbol{b}) \cdot \boldsymbol{c}

and verify that they are equal. Is the set

\{\boldsymbol{a}, \boldsymbol{b}, \boldsymbol{c}\}

right- or left-handed? (vi) By evaluating each side, verify the identity

\boldsymbol{a}\times(\boldsymbol{b}\times\boldsymbol{c}) = (\boldsymbol{a} \cdot \boldsymbol{c})\boldsymbol{b} - (\boldsymbol{a} \cdot \boldsymbol{b})\boldsymbol{c}