A vector is a force that has both magnitude and direction



We use PQ to denote the vector with initial point P and terminal point Q.

A vector is a force that has both magnitude and direction.

The direction is indicated by the arrow at the terminal point.

The magnitude is the length of the segment representing the vector. || v ||

[pic] (magnitude of the vector v) represents the length of the vector.

A vector that represents a pull or push of some type is a force vector. Examples of force vectors include a car (magnitude is speed and direction is obvious) and a thrown ball (force would be determined by size or weight of ball and speed with which it was thrown).

A single force that represents the combined forces of two combined vectors is a resultant force. An example of a resultant force would be one car hitting another car. Each car has a force. Where the hit car ends up after being hit by the second car is a result of its own force and the force of the car that hit it. We use a parallelogram to determine the resultant force. The diagonal of the parallelogram represents the resultant force.

Parallelograms are reviewed on the here and the picture is shown on the next page. Adjacent angles of a parallelogram are supplementary, opposite angles are congruent. Opposite sides are parallel and congruent. Diagonals will not be congruent.

[pic]

Vectors can be doubled, tripled, halved, etc. This new vector could be represented by multiplying the original vector by a number m. If a vector is multiplied by 2, the result is a vector in the same direction, but twice the length. If a vector is multiplied by -2, the result is a vector in the opposite direction and twice the length.

mv is a scalar multiple of the vector v. If m > 0 then it has the same direction as v.

If m < 0 then it has the opposite direction as v.

By placing a vector’s initial point at the origin, the xy-plane is used to represent the vector. The numbers a1 and a2 are the components of vector < a1, a2 >.

The direction of the vector is determined by graphing a1 and a2.

The magnitude of the vector is the length of the segment.

Magnitude of a vector: [pic]

Addition of vectors: [pic]

Subtraction of vectors: [pic]

Scalar multiple of a vector: [pic]

Ex 1) Find the magnitude of the vector [pic]

Ex 2) Find the addition vector and subtraction vectors below.

[pic]

Ex 3) Find the components of the following vectors: [pic]

A unit vector is any vector with a length 1 unit. However, there are two special unit vectors that are 1 unit long from the origin on the positive x-axis direction and on the positive y-axis direction.

Special vectors [pic] are unit vectors of magnitude 1.

Using the special unit vectors above leads to a second way to denote vectors.

An alternate way of denoting vectors: [pic]

Ex 4) Write each vector in the alternate way.

[pic]

Ex 5) Sketch vectors a and b, then find and sketch 2a, – b, a + b, a – b, and 3a + 2b

[pic] [pic]

Note: The vector a + b is the diagonal (resultant force) of a parallelogram formed by vectors a and b.

Find the magnitude of a and the smallest positive angle ( from the positive x-axis to the vector OP that corresponds to a.

[pic]

[pic]

The vectors a and b represent two forces acting at the same point, and ( is the smallest positive angle between a and b. Approximate the magnitude of the resultant force.

a = 40 lb, b = 70 lb ( = 45(

[pic]

a = 30 kg, b = 50 kg, ( = 150(

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P

Q

Every vector has 2 components; a direction and a magnitude, which is represented by the length from P to the arrow.

v

The name of this vector is v.

These are not absolute value bars.

a

b

r

Vectors a and b combined together make a resultant vector r represented by the diagonal of the parallelogram formed.

[pic]

d

Vector d shown would have the components [pic]. Components is comparable to an ordered pair.

The magnitude of vector d could be found by using the Pythagorean Thm.

When adding vectors the result would be the diagonal of the parallelogram resulting from the two given vectors.

The first would be a vector 4 times as long in the same direction. The second, a vector 3 times as long in the opposite direction.

The i here is not the imaginary unit.

a + b =

a – b =

3a + 2b =

2a – b =

a + b =

a – b =

3a + 2b =

2a – b =

[pic]

[pic]

[pic]

[pic]

[pic]

[pic]

[pic]

[pic]

[pic]

We will use the law of cosines. However, we need to complete the parallelogram to find the angle.

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