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education/math/MATH1060 (trig)/Vectors.md
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@@ -38,13 +38,13 @@ $$ \vec{v} = \langle \cos \theta,\ |\vec{v}|\sin\theta \rangle $$
# The Dot Product
The dot product of two vectors $\vec{u} = \langle a, b \rangle$ and $\vec{v} = \langle c, d \rangle$ is $\vec{u} * \vec{v} = ac + bd$.
- Given that $\vec{u} = \langle -7, 3 \rangle$, and $\vec{v} = \langle -3, 4 \rangle$, find $\vec{u} * \vec{v}$.
- $\vec{u} * \vec{v} = -7 * -4 + 3 * 4$
- Given that $\vec{u} = \langle -7, 3 \rangle$, and $\vec{v} = \langle -3, 4 \rangle$, find $\vec{u} \cdot \vec{v}$.
- $\vec{u} \cdot \vec{v} = -7 \cdot -4 + 3 \cdot 4$
The dot product can be used to find the angle between two vectors.
If $\theta (0\degree < \theta < 180\degree)$, is the angle between two nonzero vectors $\vec{u}$ and $\vec{v}$, then
$$ \cos\theta = \dfrac{\vec{u}*\vec{v}}{|\vec{u}||\vec{v}|} $$
$$ \cos\theta = \dfrac{\vec{u}\cdot\vec{v}}{|\vec{u}||\vec{v}|} $$
# Work
The dot product can be used to compute the work required to move an object a certain distance.