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Let $\text{v}=\u27e8\mathrm{-1},\mathrm{-1},1\u27e9$ and $\text{w}=\u27e82,0,1\u27e9.$ Find a unit vector in the direction of $5\text{v}+3\text{w}.$
$\u27e8\frac{1}{3\sqrt{10}},-\frac{5}{3\sqrt{10}},\frac{8}{3\sqrt{10}}\u27e9$
A quarterback is standing on the football field preparing to throw a pass. His receiver is standing 20 yd down the field and 15 yd to the quarterback’s left. The quarterback throws the ball at a velocity of 60 mph toward the receiver at an upward angle of $30\text{\xb0}$ (see the following figure). Write the initial velocity vector of the ball, $\text{v},$ in component form.
The first thing we want to do is find a vector in the same direction as the velocity vector of the ball. We then scale the vector appropriately so that it has the right magnitude. Consider the vector $\text{w}$ extending from the quarterback’s arm to a point directly above the receiver’s head at an angle of $30\text{\xb0}$ (see the following figure). This vector would have the same direction as $\text{v},$ but it may not have the right magnitude.
The receiver is 20 yd down the field and 15 yd to the quarterback’s left. Therefore, the straight-line distance from the quarterback to the receiver is
We have $\frac{25}{\Vert \text{w}\Vert}=\text{cos}\phantom{\rule{0.2em}{0ex}}30\text{\xb0}.$ Then the magnitude of $\text{w}$ is given by
and the vertical distance from the receiver to the terminal point of $\text{w}$ is
Then $\text{w}=\u27e820,15,\frac{25}{\sqrt{3}}\u27e9,$ and has the same direction as $\text{v}.$
Recall, though, that we calculated the magnitude of $\text{w}$ to be $\Vert \text{w}\Vert =\frac{50}{\sqrt{3}},$ and $\text{v}$ has magnitude $60$ mph. So, we need to multiply vector $\text{w}$ by an appropriate constant, $k.$ We want to find a value of $k$ so that $\Vert k\text{w}\Vert =60$ mph. We have
so we want
Then
Let’s double-check that $\Vert \text{v}\Vert =60.$ We have
So, we have found the correct components for $\text{v}.$
Assume the quarterback and the receiver are in the same place as in the previous example. This time, however, the quarterback throws the ball at velocity of $40$ mph and an angle of $45\text{\xb0}.$ Write the initial velocity vector of the ball, $\text{v},$ in component form.
$\text{v}=\u27e816\sqrt{2},12\sqrt{2},20\sqrt{2}\u27e9$
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