Vector Algebra

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Section 1.4 Vector Algebra

Representing vectors as arrows is a good way to get a geometric understanding and also for creating a good physics diagram of a physical situation you are interested in understanding. However, if you want to quantify the physics, you need to work algebraically within a particular coordinate system. Consider the vector

\vec{A}

written in terms of components and the unit vectors

\hat{x},

\hat{y},

and

\hat{z}

\vec{A} = a_{x} \hat{x} + a_{y} \hat{y} + a_{z} \hat{z}

Now you can algebraically compute the vector operations you learned previously! For example, to multiply a vector by a scalar algebraically, you multiply each component by the scalar separately.

a \vec{A} = (a A_{x}) \hat{x} + (a A_{y}) \hat{y} + (a A_{z}) \hat{z}

Key Skill 1.4.1. Vector Addition and Subtraction.

To add vectors that are written in terms of components algebraically, you add the components (

x,

y,

and

z,

for example) for the two vectors separately.

\begin{aligned} \vec{A} + \vec{B} & = (A_{x} \hat{x} + A_{y} \hat{y} + A_{z} \hat{z}) + (B_{x} \hat{x} + B_{y} \hat{y} + B_{z} \hat{z}) \\ = (A_{x} + B_{x}) \hat{x} + (A_{y} + B_{y}) \hat{y} + (A_{z} + B_{z}) \hat{z} \end{aligned}

Similarly, to subtract vectors algebraically, you subtract like components.

\begin{aligned} \vec{A} - \vec{B} & = (A_{x} \hat{x} + A_{y} \hat{y} + A_{z} \hat{z}) - (B_{x} \hat{x} + B_{y} \hat{y} + B_{z} \hat{z}) \\ = (A_{x} - B_{x}) \hat{x} + (A_{y} - B_{y}) \hat{y} + (A_{z} - B_{z}) \hat{z} \end{aligned}

Exercises Practice Activities

Use the following two vectors for these activities:

\vec{A} = - 5 \hat{x} - \hat{y}

and

\vec{B} = - 2 \hat{x} + 3 \hat{y}

1.

Find

- \vec{B} / 5 .

- \vec{B} / 5 = \hat{x} + \frac{1}{5} \hat{y}

2.

Find

\vec{A} + 2 \vec{B} .

\vec{A} + 2 \vec{B} = - 9 \hat{x} + 5 \hat{y}

3.

Find

\vec{A} - \vec{B} .

\vec{A} - \vec{B} = - 3 \hat{x} - 4 \hat{y}

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