Vectors - the Basics

Vectors - the Basics - How it Works - Video

Vectors - What are they?

Vectors - What are they?:

Vectors have a magnitude and direction. You can think of a vector as a directed line segment. The magnitude is the length of that segment or vector with an arrow pointed in the direction.

The initial point or the tail is where the vector starts and the terminal or the head is where the vector ends.

Vectors - How do we write them?

Vectors - How do we write them?:

One way to write them is with one lowercase bold letter. Sometimes you might given an initial and terminal point. You write the initial point first then the terminal point and arrow on top of the letters. The third way is with i and j. Why those letters? I am not sure, but i goes horizontally (x-axis) and j goes vertically (y-axis).

Vectors - Definitions

Vectors - Definitions:

The magnitude helps us find the distance of our vector.

We have 3 special vectors the 0 vector the i vector and j vector. The 0 vector is <0, 0>; the i is <1, 0>; the j is <0, 1>.

Example 1

Example 1:

The magnitude helps us find the distance of our vector.

We have 3 special vectors the 0 vector the i vector and j vector. The 0 vector is <0, 0>; the i is <1, 0>; the j is <0, 1>.

Example 1:

We have our vector m = <6, 8>. We graph the vector just like any coordinate point. We use the magnitude formula to find distance.

We could like at the picture at the right. We have formed a right triangle. What does the red line represent? Does the magnitude formula look familiar? It is the Pythagorean Theorem but one step in. So each time we find the magnitude of a vector we are basically finding the hypotenuse of a right triangle.

Example 2

Example 2:

We have our vector m = <3, 4> and t = <2, 1>. Here we are adding our vectors. When we add vectors, it is just like regular addition. When we add, we add the corresponding parts, so the first parts in each vector, then the second parts in each vector, and so on...

So the resultant vector is <5, 1>.

The next question that you might ask is, Does order matter? What is 2 + 3? What is 1 + 4?

One of the differences, when we add this way t + m, is on the graph. The t vector goes first then the m vector goes second, which forms a parallelogram and magnitude cuts that kite in parallelogram.

Here we can see what happens on the graph when we switch the order when adding.

Example 3

Example 3:

We have our vectors m = <3, 4> and z = <2, 4>. Here we are subtracting our vectors. When we subtract vectors, it is just like regular subtraction. When we subtract, we subtract the corresponding parts,so the first parts in each vector, then the second parts in each vector, and so on...

So the resultant vector is <1, 0>.

Does this vector look like something that we have talked about? It is the i vector.

On the graph the m vector is going in original direction 1 times and then we subtract the z vector in its opposite direction 1 time. After all of that we end up at <8, 1>.

Example 4

Example 4:

We have our vectors m = <5, 4> and t = <2, 1>. Here we are scaling (multiplication) each vector then subtracting. When we scale vectors, it is just like regular multiplication. When we scale, we multiply each number in the vector by the number out in front.

So the resultant vector is <2, 4>.

In our graph we went m and then added another m so we have 2m. Next we subtracted t four times or 4t.

On the graph the m vector is going in original direction 2 times and then we subtract the t vector in its opposite direction 4 times. So we are going <-2, -1> four times. After all of that we end up at <8, 1>.

Example 5

Example 5:

We have our vectors t = <2, 1> and z = <-2, -4>. Here we are scaling (multiplication) each vector then adding. When we scale vectors, it is just like regular multiplication. When we scale, we multiply each number in the vector by the number out in front.

So the resultant vector is <8, 1>.

In our graph we went t and then added t four more times so we have 5t. Next we add z one time.

On the graph the t vector is going in original direction 5 times and then we add the z vector in its original direction 1 time. After all of that we end up at <8, 1>.

Live Worksheet - Add and Subtract Vectors

Live Worksheet - Scalar Multiples

Live Worksheet - Magnitude

Teacher - Edpuzzle Link