1.
Find \(-\vec{B}/5\text{.}\)
Answer.
\(-\vec{B}/5 = \hat{x}+\frac{1}{5}\hat{y}\)
Section 1.5 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 \(\hat{x} \text{,}\) \(\hat{y} \text{,}\) and \(\hat{z} \)
\begin{equation*}
\vec{A} = a_x\hat{x}+a_y\hat{y}+a_z\hat{z}
\end{equation*}
Now you can algebraically compute the vector operations you learned previously!
Exercises Vector Addition
To add vectors algebraically, you simply add like components.
\begin{align*}
\vec{A} + \vec{B} \amp = (A_x \hat{x} + A_y \hat{y}+ A_z\hat{z}) + (B_x \hat{x} + B_y \hat{y}+ B_z\hat{z}) \\
\amp = (A_x +B_x)\hat{x} + (A_y + B_y)\hat{y} + (A_z+B_z) \hat{z}
\end{align*}
Exercises Vector Subtraction
To subtract vectors algebraically, you simple subtract like components.
\begin{align*}
\vec{A} - \vec{B} \amp = (A_x \hat{x} + A_y \hat{y}+ A_z\hat{z}) - (B_x \hat{x} + B_y \hat{y}+ B_z\hat{z}) \\
\amp = (A_x - B_x)\hat{x} + (A_y - B_y)\hat{y} + (A_z - B_z) \hat{z}
\end{align*}
Exercises Multiplication of a vector by a scalar
To multiply a vector by a scalar algebraically you multiply each component.
\begin{equation*}
a\vec{A} = (aA_x)\hat{x} + (aA_y)\hat{y} + (aA_z)\hat{z}
\end{equation*}
Exercises The dot product of two vectors
To compute the dot product algebraically, you multiply like components and add them together.
\begin{align*}
\vec{A} \cdot \vec{B} \amp = (A_x \hat{x} + A_y \hat{y}+ A_z\hat{z}) \cdot (B_x \hat{x} + B_y \hat{y}+ B_z\hat{z})\\
\amp = A_xB_x(\hat{x} \cdot \hat{x}) + A_yB_y(\hat{y} \cdot \hat{y}) + A_zB_z (\hat{z} \cdot \hat{z}) \\
\amp = A_xB_x + A_yB_y+A_zB_z
\end{align*}
\(\hat{x} \cdot \hat{x} = \hat{y} \cdot \hat{y} = \hat{z} \cdot \hat{z} = 1\)Exercises Practice Activities
Use the following two vectors for these activities: \(\vec{A}=-5\hat{x}-\hat{y}\) and \(\vec{B}=-2\hat{x}+3\hat{y} \)
2.
Find \(\vec{A} + 2\vec{B}\text{.}\)
Answer.
\(\vec{A} + 2\vec{B} = -9\hat{x}+5\hat{y} \)
3.
Find \(\vec{A} - \vec{B}\text{.}\)
Answer.
\(\vec{A} - \vec{B} = -3\hat{x}-4\hat{y}\)
