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. \(1 \mathbf{a} =\mathbf{a}\)
$$
\mathbf{i} = \left \langle 1, 0, 0 \right \rangle \; \; \mathbf{j} = \left \langle 0, 1, 0 \right \rangle \; \; \mathbf{k} = \left \langle 0, 0, 1 \right \rangle
$$

E 13.2 Exercises