Question

Amy is pulling a wagon with a force of 30 pounds up a hill at an angle of 25°. Give the force exerted on the wagon as a vector and as a linear combination of unit vectors i and j.

Answer 1

Vector (ordered pair - rectangular form)

`\vec F = (27.189,12.679)\,[lbf]`

Vector (ordered pair - polar form)

`\vec F = (30\,lbf, 25^(\circ))`

Sum of vectorial components (linear combination)

`\vec F = 27.189\cdot \hat{i} + 12.679\cdot \hat{j}\,[N]`

Explanation:

From statement we know that force exerted on the wagon has a magnitude of 30 pounds-force and an angle of 25° above the horizontal, which corresponds to the +x semiaxis, whereas the vertical is represented by the +y semiaxis.

The force (
`\vec F`), in pounds-force, can be modelled in two forms:

Vector (ordered pair - rectangular form)

`\vec F = \left(\|\vec F\|\cdot \cos \theta, \|\vec F\|\cdot \sin \theta\right)` (1)

Vector (ordered pair - polar form)

`\vec F = \left(\|\vec F\|, \theta\right)`

Sum of vectorial components (linear combination)

`\vec {F} = \left(\|\vec F\|\cdot \cos \theta\right)\cdot \hat{i} + \left(\|\vec F\|\cdot \sin \theta \right)\cdot \hat{j}` (2)

Where:

`\|\vec F\|` - Norm of the vector force, in newtons.

`\theta` - Direction of the vector force with regard to the horizontal, in sexagesimal degrees.

`\hat{i}`,
`\hat{j}` - Orthogonal axes, no unit.

If we know that
`\|\vec F\| = 30\,lbf` and
`\theta = 25^(\circ)`, then the force exerted on the wagon is:

Vector (ordered pair - rectangular form)

`\vec F= \left(30\cdot \cos 25^(\circ), 30\cdot \sin 25^(\circ)\right)\,[lbf]`

`\vec F = (27.189,12.679)\,[lbf]`

Vector (ordered pair - polar form)

`\vec F = (30\,lbf, 25^(\circ))`

Sum of vectorial components (linear combination)

`\vec F = (30\cdot \cos 25^(\circ))\cdot \hat{i} + (30\cdot \sin 25^(\circ))\cdot \hat{j}\,[N]`

`\vec F = 27.189\cdot \hat{i} + 12.679\cdot \hat{j}\,[N]`