2 Vectors
2.3 Algebra of Vectors
Learning Objectives
By the end of this section, you will be able to:
- Apply analytical methods of vector algebra to find resultant vectors and to solve vector equations for unknown vectors.
- Interpret physical situations in terms of vector expressions.
Vectors can be added together and multiplied by scalars. Vector addition is associative (Figure) and commutative (Figure), and vector multiplication by a sum of scalars is distributive (Figure). Also, scalar multiplication by a sum of vectors is distributive:
In this equation, [latex]\alpha[/latex] is any number (a scalar). For example, a vector antiparallel to vector [latex]\mathbf{\overset{\to }{A}}={A}_{x}\mathbf{\hat{i}}+{A}_{y}\mathbf{\hat{j}}+{A}_{z}\mathbf{\hat{k}}[/latex] can be expressed simply by multiplying [latex]\mathbf{\overset{\to }{A}}[/latex] by the scalar [latex]\alpha =-1[/latex]:
Example
Direction of Motion
In a Cartesian coordinate system where [latex]\mathbf{\hat{i}}[/latex] denotes geographic east, [latex]\mathbf{\hat{j}}[/latex] denotes geographic north, and [latex]\mathbf{\hat{k}}[/latex] denotes altitude above sea level, a military convoy advances its position through unknown territory with velocity [latex]\mathbf{\overset{\to }{v}}=(4.0\mathbf{\hat{i}}+3.0\mathbf{\hat{j}}+0.1\mathbf{\hat{k}})\text{km}\text{/}\text{h}[/latex]. If the convoy had to retreat, in what geographic direction would it be moving?
Solution
Show Answer
The velocity vector has the third component [latex]{\mathbf{\overset{\to }{v}}}_{z}=(+0.1\text{km}\text{/}\text{h})\mathbf{\hat{k}}[/latex], which says the convoy is climbing at a rate of 100 m/h through mountainous terrain. At the same time, its velocity is 4.0 km/h to the east and 3.0 km/h to the north, so it moves on the ground in direction [latex]{\text{tan}}^{-1}(3\,\text{/}4)\approx 37^\circ[/latex] north of east. If the convoy had to retreat, its new velocity vector [latex]\mathbf{\overset{\to }{u}}[/latex] would have to be antiparallel to [latex]\mathbf{\overset{\to }{v}}[/latex] and be in the form [latex]\mathbf{\overset{\to }{u}}=\text{−}\alpha \mathbf{\overset{\to }{v}}[/latex], where [latex]\alpha[/latex] is a positive number. Thus, the velocity of the retreat would be [latex]\mathbf{\overset{\to }{u}}=\alpha (-4.0\mathbf{\hat{i}}-3.0\mathbf{\hat{j}}-0.1\mathbf{\hat{k}})\text{km}\text{/}\text{h}[/latex]. The negative sign of the third component indicates the convoy would be descending. The direction angle of the retreat velocity is [latex]{\text{tan}}^{-1}(-3\alpha \text{/}-4\alpha )\approx 37^\circ[/latex] south of west. Therefore, the convoy would be moving on the ground in direction [latex]37^\circ[/latex] south of west while descending on its way back.
The generalization of the number zero to vector algebra is called the null vector, denoted by [latex]\mathbf{\overset{\to }{0}}[/latex]. All components of the null vector are zero, [latex]\mathbf{\overset{\to }{0}}=0\mathbf{\hat{i}}+0\mathbf{\hat{j}}+0\m