Question

Find a vector perpendicular to the line 2y=5-3x in the xy-plane.

Answer 1

Solution: Given an equation of a line:

The objective is to find a vector perpendicular to this line in the xy-plane i.e. the z-component of the vector must be zero.

Comparing the given equation with the slope-intercept form of the line, we'll get:

Therefore, the slope of the given line is,

The relation between the slopes of two perpendicular line is,

...... (1)

Therefore, the slope of the line perpendicular to the given line is,

Therefore, the equation of line perpendicular to the given line is of the form:

Now, for any value of the constant c, this line will be perpendicular to the given line because its slope is satisfying the relation given by equation (1), therefore, for simplicity, we'll assume c to be 0. Thus, the equation of the line will become:

...... (2)

Now, we'll find any two points on the above line. For x = 0, we'll get y = 0. Thus, we get the point (0,0). For x = 3, we'll get y = 2, therefore, the point (3,2) lies on this line.

Now, a vector perpendicular to the given line must lie along the line (2). The vector between two points is given by:

Therefore,

This is one of the vectors which is perpendicular to the given line as for different values of constant c and points on the line the vector may vary.

I hope it helps you!