Points, Vectors, and Functions Exercises

Answer

Draw the triangle for the point

We know

therefore

To find r we use the Pythagorean Theorem:

We conclude that the polar coordinates of this point are

Now draw the triangle for the point (1, 1):

We have

and

The polar coordinates of the point are

Draw the triangle for the point

We have

therefore

Using the Pythagorean theorem,

The polar coordinates of the point are

Answer

Draw the triangle for the point :

We have

therefore

For y we have

therefore

The rectangular coordinates of the point are

.

Now draw the triangle for the point :

Since the x and y coordinates will be the same, which means we only need to find one of them.

Therefore

.

The rectangular coordinates of the point are

.

Lastly, the point is on an axis, so we don't need a triangle.

We can see that the x coordinate is 0 and the y coordinate is 7. The rectangular coordinates of this point are (x, y) = (0, 7).

Answer

To translate into polar coordinates we need to find the magnitude r and direction θ of the vector with the given components.This point, or vector, is in the third quadrant. The magnitude is

The direction is .

The polar coordinates are

These coordinates describe a point in the third quadrant, this is a reasonable answer.

This point is in the second quadrant. The magnitude is

The direction is .

The polar coordinates are

These coordinates still describe a point in the second quadrant, this is a reasonable answer.

This point is in the fourth quadrant. The magnitude is

The direction is . 

The polar coordinates are

This describes a point in the fourth quadrant, this is a reasonable answer.

The point (0, -5) lies on the negative y-axis.We can see that its magnitude is 5 and its direction is 3π/2, its polar coordinates are .

Answer

• To translate into rectangular coordinates we apply the formulas x = r cos θ and y = r sin θ.

• The point is in the second quadrant.

The rectangular coordinates are

which do indeed describe a point in the second quadrant.

• The point lies in the third quadrant.

The rectangular coordinates are

which do describe a point in the third quadrant.

• The point is in the third quadrant.

The rectangular coordinates are

which are indeed in the third quadrant.

• The point (4, π) is on the negative x-axis.

• We don't need to bother with formulas because we can see immediately that the rectangular coordinates of this point are (x, y) = (-4, 0).