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Vectors and Dot Products
Vectors and Dot Products
A vector has magnitude and direction. In coordinates, $$\mathbf{v}=\langle a,b \rangle$$ in 2D or $$\langle a,b,c \rangle$$ in 3D.
Basic Operations
Add vectors component-wise and multiply by scalars component-wise.
Example 1
Add $\langle 2,-1 \rangle + \langle 4,3 \rangle$.
$\langle 6,2 \rangle$.
Magnitude
$$\|\mathbf{v}\|=\sqrt{a^2+b^2}$$ in 2D.
Example 2
Find the magnitude of $\langle 3,4 \rangle$.
$5$.
Dot Product
$$\mathbf{u}\cdot\mathbf{v}=u_1v_1+u_2v_2$$
If the dot product is $0$, the vectors are perpendicular.
Example 3
Compute $\langle 1,2 \rangle\cdot\langle 3,4 \rangle$.
$1(3)+2(4)=11$.
Practice Problems
1. Add $\langle 1,5 \rangle$ and $\langle -2,4 \rangle$.
2. Find the magnitude of $\langle 6,8 \rangle$.
3. Compute $\langle 2,1 \rangle\cdot\langle 5,-3 \rangle$.
4. What does a dot product of zero mean?
5. Multiply $2\langle 3,-1 \rangle$.
Show Answer Key
1. $\langle -1,9 \rangle$
2. $10$
3. $7$
4. The vectors are perpendicular
5. $\langle 6,-2 \rangle$