# 13.2 | Vectors in General

Given the points $$A(x_1, y_1, z_1)$$ and $$B(x_2, y_2, z_2)$$, the vector $$\mathbf{a}$$ with representation $$\vec{AB}$$ is
$$\mathbf{a} = \left \langle x_2 – x_1, y_2 – y_1, z_2 – z_1 \right \rangle$$
The length of the two-dimensional vector $$\mathbf{a} = \left \langle a_1, a_2 \right \rangle$$ is
$$\left| \mathbf{a} \right| = \sqrt{ a_1^2 + a_2^2 }$$
The length of the three-dimesional vector $$\mathbf{a} = \left \langle a_1, a_2, a_3 \right \rangle$$
$$\left| \mathbf{a} \right| = \sqrt{ a_1^2 + a_2^2 + a_3^2}$$
If $$\mathbf{a} = \left \langle a_1, a_2 \right \rangle$$ and $$\mathbf{b} = \left \langle b_1, b_2 \right \rangle$$, then

• $$\mathbf{a + b} = \left \langle a_1+b_1, a_2+b_2 \right \rangle$$
• $$\mathbf{a – b} = \left \langle a_1-b_1, a_2-b_2 \right \rangle$$
• $$c\mathbf{a} = \left \langle ca_1, ca_2 \right \rangle$$

Similarly, for three-dimensional vectors,
\begin{aligned} \left \langle a_1, a_2, a_3 \right \rangle+\left \langle b_1, b_2, b_3 \right \rangle & = \left \langle a_1+b_1, a_2+b_2, a_3+b_3 \right \rangle \\ \left \langle a_1, a_2, a_3 \right \rangle-\left \langle b_1, b_2, b_3 \right \rangle & = \left \langle a_1-b_1, a_2-b_2, a_3-b_3 \right \rangle \\ c\left \langle ca_1, ca_2,ca_3 \right \rangle &= \left \langle ca_1, ca_2, ca_3 \right \rangle \end{aligned}

Properties of Vectors: If $$\mathbf{a}, \mathbf{b},$$ and $$\mathbf{c}$$ are vectors in $$V_n$$ and $$c$$ and $$d$$ are scalars, then

1. $$\mathbf{a} + \mathbf{b} = \mathbf{b} + \mathbf{a}$$
2. $$\mathbf{a} + \mathbf{0} = \mathbf{a}$$
3. $$c(\mathbf{a} + \mathbf{b}) = c\mathbf{b} + c\mathbf{a}$$
4. $$(cd)\mathbf{a} = c(d\mathbf{a})$$
5. $$\mathbf{a} + (\mathbf{b}+ \mathbf{c}) = (\mathbf{a} + \mathbf{b})+ \mathbf{c}$$
6. $$\mathbf{a}+ (-\mathbf{a}) = \mathbf{0}$$
7. $$(c+d)\mathbf{a} = c\mathbf{a}+ d\mathbf{a}$$
8. $$1 \mathbf{a} =\mathbf{a}$$
$$\mathbf{i} = \left \langle 1, 0, 0 \right \rangle \; \; \mathbf{j} = \left \langle 0, 1, 0 \right \rangle \; \; \mathbf{k} = \left \langle 0, 0, 1 \right \rangle$$