Q. 10 - Calculus

Question

How do you subtract two vectors algebraically? Geometrically 

Step-by-Step Solution

Answer

$$\begin{gathered} u=x_{1}i+y_{1}j \mathrm{and} v=x_{2}i+y_{2}j \\ \overrightarrow{u}-\overrightarrow{v}=\left(x_1-x_2\right)i+(y_1-y_2) \end{gathered}$$

Step 1: Subtract two vectors algebraically

Let two vectors are,

$$\begin{gathered} u=x_{1}i+y_{1}j \mathrm{and} v=x_{2}i+y_{2}j \\ \overrightarrow{u}-\overrightarrow{v}=\left(x_1-x_2\right)i+(y_1-y_2) \end{gathered}$$

Example:

$$\begin{gathered} \overrightarrow{u}=3i+8j \mathrm{and} \overrightarrow{v}=2i-4j \\ \overrightarrow{u}-\overrightarrow{v}=\left(3i-2i\right)+\left(8j+4j\right) \\ =i+12j \end{gathered}$$

Step 2: Subtract two vectors geometrically

To subtract two vectors geometrically we position their initial points together, as we see in the right-hand figure. The vector $u - v$ extends from the terminal point of v to the terminal point of u.

Example:

$$\begin{gathered} \overrightarrow{u}=3i+8j \mathrm{and} \overrightarrow{v}=2i-4j \\ \overrightarrow{u}-\overrightarrow{v}=\left(3i-2i\right)+\left(8j+4j\right) \\ =i+12j \end{gathered}$$