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 \(\hat{x} \text{,}\) \(\hat{y} \text{,}\) 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! For example, to multiply a vector by a scalar algebraically, you multiply each component by the scalar separately.
\begin{equation*}
a\vec{A} = (aA_x)\hat{x} + (aA_y)\hat{y} + (aA_z)\hat{z}
\end{equation*}
Key Skill 1.4.1. Vector Addition and Subtraction.
To add vectors that are written in terms of components algebraically, you add the components (\(x\text{,}\) \(y\text{,}\) and \(z\text{,}\) for example) for the two vectors separately.
\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*}
Similarly, to subtract vectors algebraically, you 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 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 \(-\vec{B}/5\text{.}\)
Answer.\(-\vec{B}/5 = \hat{x}+\frac{1}{5}\hat{y}\)
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.\(\vec{A} - \vec{B} = -3\hat{x}-4\hat{y}\)
