# Precalculus : Algebraic Vectors and Parametric Equations

## Example Questions

### Example Question #8 : Express A Vector In Polar Form

Write the vector in polar form, .

Explanation:

It will be helpful to first draw the vector so we can see what quadrant the angle is in:

Since the vector is pointing up and to the right, it is in the first quadrant. To determine the angle, set up a trig equation with tangent, since the component 5 is opposite and the component 4 is adjacent to the angle we are looking for:

to solve for theta, take the inverse tangent of both sides:

Now we have the direction, and we can solve for the magnitude using Pythagorean Theorem:

take the square root of both sides

The vector in polar form is

### Example Question #1 : Express A Vector In Component Form

Express the following vector in component form:

Explanation:

When separating a vector into its component form, we are essentially creating a right triangle with the vector being the hypotenuse.

Therefore, we can find each component using the cos (for the x component) and sin (for the y component) functions:

We can now represent these two components together using the denotations i (for the x component) and j (for the y component).

### Example Question #1 : Express A Vector In Component Form

Find , then find its magnitude.  and  are both vectors.

Explanation:

In vector addition, you simply add each component of the vectors to each other.

x component: .

y component: .

z component: .

The new vector is

.

To find the magnitude we use the formula,

Thus its magnitude is 5.

### Example Question #1 : Express A Vector In Component Form

Find the component form of the vector with

initial point

and

terminal point .

Explanation:

To find the vector in component form given the initial and terminal points, simply subtract the initial point from the terminal point.

### Example Question #4 : Express A Vector In Component Form

Find the component form of the vector with

initial point

and

terminal point

Explanation:

To find the vector in component form given the initial and terminal points, simply subtract the initial point from the terminal point.

### Example Question #5 : Express A Vector In Component Form

A bird flies 15 mph up at an angle of 45 degrees to the horizontal.  What is the bird's velocity in component form?

Explanation:

Write the formula to find both the x and y-components of a vector.

Substitute the value of velocity and theta into the equations.

The vector is:

### Example Question #6 : Express A Vector In Component Form

Write this vector in component form:

Explanation:

In order to find the horizontal component, set up an equation involving cosine with 7 as the hypotenuse, since the side in the implied triangle that represents the horizontal component is adjacent to the 22-degree angle:

First, find the cosine of 22, then multiply by 7

To find the vertical component, set up an equation involving sine, since the side in the implied triangle that represents the vertical component is opposite the 22-degree angle:

First, find the sine of 22, then multiply by 7

We are almost done, but we need to make a small adjustment. The picture indicates that the vector points up and to the left, so the horizontal component, 6.49, should be negative:

### Example Question #121 : Matrices And Vectors

Write a vector equation describing the line passing through P1 (1, 4) and parallel to the vector  = (3, 4).

Explanation:

First, draw the vector  = (3, 4); this is represented in red below. Then, plot the point P1 (1, 4), and draw a line  (represented in blue) through it that is parallel to the vector .

We must find the equation of line . For any point P2 (x, y) on . Since  is on line  and is parallel to  for some value of t. By substitution, we have . Therefore, the equation  is a vector equation describing all of the points (x, y) on line  parallel to  through P1 (1, 4).

### Example Question #122 : Matrices And Vectors

True or false: A line through P1 (x1, y1) that is parallel to the vector  is defined by the set of points  such that  for some real number t. Therefore, .

False

True

True

Explanation:

This is true. The independent variable  in this equation is called a parameter.

### Example Question #123 : Matrices And Vectors

Find the parametric equations for a line parallel to and passing through the point (0, 5).

x = 5t

y = 3 + 2t

x = 3

y = 2 + 5t

x = 3 + 2t

y = 5t

x = 3t

y = 5 + 2t

x = 3t

y = 5 + 2t

Explanation:

A line through a point (x1,y1) that is parallel to the vector  = (a1, a2) has the following parametric equations, where t is any real number.

Using the given vector and point, we get the following:

x = 3t

y = 5 + 2t

Each value of t creates a distinct (x, y) ordered pair. You can think of these points as representing positions of an object, and of t as representing time in seconds. Evaluating the parametric equations for a value of t gives us the coordinates of the position of the object after t seconds have passed.