1.
-
- Find the height [latex]h[/latex] of the triangle.
- How far is the foot of the altitude from the vertex of the [latex]18°[/latex] angle?
2.
- Find the height [latex]h[/latex] of the triangle.
- Find the length of the third side of the triangle.
3.
- How far north is the tower from the airport? How far east?
- What is the distance from [latex]A[/latex] to [latex]P\text{?}[/latex]
4.
- Find the distance from [latex]A[/latex] to [latex]C\text{.}[/latex]
- How far north is point [latex]C[/latex] from point [latex]B\text{?}[/latex]
[latex]\underline{\qquad\qquad\qquad\qquad}[/latex]
Skills Refresher Answers
1.
- [latex]\displaystyle 1.55[/latex]
- [latex]\displaystyle 4.76[/latex]
2.
- [latex]\displaystyle 19.57[/latex]
- [latex]\displaystyle 81.83[/latex]
3.
- 36.92 mi, 15.63 mi
- 16.08 mi
4.
- [latex]\displaystyle 21.87[/latex]
- [latex]\displaystyle 5.95[/latex]
- Find the component of [latex]{\bf{w}}[/latex] in the direction of [latex]{\bf{v}}[/latex].
- Compute the dot product.
- Find the angle between two vectors.
- Resolve a vector into components in given directions.
We have seen that it can be useful to resolve a vector into horizontal and vertical components. We can also break a vector into components that point in other directions.
Imagine the following experiment: Delbert holds a ball at shoulder height and then drops it so that it falls to the ground. Francine holds a ball at shoulder height on an inclined ramp, then releases it so that it rolls downhill. Which ball will reach the ground first?
Although gravity causes both balls to speed up, the free-falling ball will reach the ground first. The force of gravity pulls straight down, the same direction as the motion of the free-falling ball, but the rolling ball must move at an angle to the pull of gravity, along the surface of the ramp. Only part of the gravitational force accelerates the rolling ball, and the rest of the force is counteracted by the surface of the ramp. What fraction of the gravitational force causes the ball to roll?
In figure (a), the gravitational force [latex]{\bf{F}}[/latex] is resolved into the sum of two vectors, [latex]{\bf{F}}={\bf{u}}+{\bf{v}}\text{,}[/latex] where [latex]{\bf{v}}[/latex] points down the ramp, and [latex]{\bf{u}}[/latex] is perpendicular to the ramp. The magnitude of [latex]{\bf{v}}[/latex] is called the component of [latex]{\bf{F}}[/latex] in the direction of motion, and is denoted by [latex]\text{comp}_{\bf{v}}{\bf{F}}\text{.}[/latex] This is the portion of the gravitational force that moves the ball. From figure (b), we see that [latex]\text{comp}_{\bf{v}}{\bf{F}} = \|{\bf{F}}\| \cos \theta\text{,}[/latex] where [latex]\theta[/latex] is the angle between [latex]{\bf{F}}[/latex] and [latex]{\bf{v}}\text{.}[/latex]
The component of a vector [latex]{\bf{F}}[/latex] in the direction of vector [latex]{\bf{v}}[/latex] is
[latex]{\text{comp}_{\bf{v}}{\bf{F}} = \|{\bf{F}}\| \cos \theta}[/latex]
where [latex]\theta[/latex] is the angle between