Physics

Problem

A trekker starts from a camp at the origin

O

and follows three successive displacements: 1.

\mathbf{u}=5\,\hat i\,\text{km}

(due east), 2.

\mathbf{v}=3\sqrt2\bigl(\tfrac{1}{\sqrt2}\,\hat i+\tfrac{1}{\sqrt2}\,\hat j\bigr)\,\text{km}

(north-east at

45^{\circ}

to the east axis), 3.

\mathbf{w}=-\tfrac12\,\mathbf{u}+\lambda\,\mathbf{v}

, where

\lambda

is a real parameter. a. (i) Write

\mathbf{w}

explicitly in terms of

\hat i

and

\hat j

for a general value of

\lambda

. b. (ii) Obtain the resultant displacement

\mathbf{R}=\mathbf{u}+\mathbf{v}+\mathbf{w}

in component form. c. (iii) Determine the value of

\lambda

for which the trekker’s net displacement

\mathbf{R}

is perpendicular to the initial displacement

\mathbf{u}

. Also find the magnitude of

\mathbf{R}

for this value of

\lambda