Figure 1 Coordinate system

Example 13: Two vectors are given by A = −3i + 4j and B = 2i + 3j. Find (a) A × B and (b) the angle between A and B.
Solution: To solve this example we can use determinant, Eq. (14). But to solve determinant we apply Eq. 13


(b) To get the angle between A and B it is easiest to use the dot product and Pythagorean theorem. The magnitudes of A and B are




and the dot product of the two vectors is

so then if is the angle between A and B we get

which gives

Summary
Cartesian coordinate system The Cartesian plane is formed by using two lines intersecting at right angles. The horizontal line is usually called the x-axis, and the vertical line is usually called the y-axis. The point of intersection of these two axes is the origin, and the two axes divide the plane into four parts called quadrants. The x-axis contains positive numbers to the right of the origin and negative numbers to the left of the origin. The y-axis contains positive numbers above the origin and negative numbers below the origin.

Component of a vector and vector addition Any vectors making an angle with horizontal or vertical directions have two components. On x-axis this component is A_x and on y-axis it is A_y. Mathematically it is

These components form two sides of a right triangle having a hypotenuse with magnitude A. It follows that A’s magnitude and direction are related to its components through the Pythagorean theorem and the definition of the tangent:


If there are two or more vectors firstly components of each vector is found. Then x-component each vector is added to find resultant x-component and y-component of each vector is added to find resultant vector on y-axis.

Through the Pythagorean theorem final resultant vector is found.