## Wiring up a Net-Zero Home

A homeowner decides to make their house carbon-neutral. They place solar panels on the roof, which then connect o their mains circuit via a wire.

The wire can be modelled as leaving the solar panels at A = 2i + 3j + 4k, and connecting to the mains at the point B = -3i + j – 3k, with the distances measured in metres and both points measured relative to the same fixed origin.

a) Show that $$\overrightarrow{\text{AB}} = – 5\mathbf{i -}2\mathbf{j -}7\mathbf{k}$$ and hence find the length of wire needed to 2 decimal places.

[2 marks]

In many homes powered by solar energy, when excess power is generated, it can be put onto the national grid, so that more renewables power the grid rather than fossil fuels.

The wire leading to the grid is on an automatic switch system M, which divides the wire $$\overrightarrow{\text{AB}}$$ in the ratio 2:1.

b) Calculate the distance of the automatic switch system M from the origin.

[4 marks]

## A Hurricane Direction Shift

The movement of a hurricane is modelled by vector h.

It is moving at a speed of 20 kph, with the direction of 165˚ above the equator, when portrayed on a flat map of the Earth.

a) Write h in component form.

[2 marks]

The hurricane makes landfall. Its movement is now modelled by vector l, $$\left( \frac{15\sqrt{3}}{2}\mathbf{\text{i}},\mathbf{\ }\frac{15}{2}\mathbf{j} \right)$$.

b) Find the amount by which the hurricanes speed has decreased and state the hurricanes new direction.

[3 marks]

## Monitoring Currents

In this question, all distances are measured in kilometres.

2 deep sea ocean current monitors, A and B, have position vectors (-1, 7, k) and (4, 1, 10) respectively, relative to a fixed origin. Given that the distance from A to B is $$5\sqrt{5}$$km,

a) Find the possible values of the constant k.

[3 marks]

b) For the larger value of k, find the unit vector in the direction of $$\overrightarrow{\text{OA}}$$

[3 marks]

## Mountainside Monitoring

3 CO₂ monitors K, L and M are placed on a mountain side

The vector $$\overrightarrow{\text{KL}} = 3\mathbf{i -}6\mathbf{k}$$ and $$\overrightarrow{\text{LM}} = 2\mathbf{i} + 5\mathbf{j} + 4\mathbf{k}$$, relative to a fixed origin.

Show that $$\angle KLM = 66.4°$$ to one decimal place

[7 marks]

Hence find $$\angle LKM$$ and $$\angle LMK$$

[3 marks]

## Colliding Currents

2 deep sea ocean currents meet.

By modelling one current as the positive y axis.

a) Find the angle that the second current, with vector $$4\mathbf{i – j}$$, makes with the first current when they meet.

[3 marks]