vector

Examples of Vectors

Examples of Scalars

Examples of Polar Vectors

Polar form always states the magnitude first and then the direction. In physics, the direction is usually from the positive x axis and rotates counterclockwise following trig notation.

Examples of Component Vectors

Two types of component notation are common.

1. Using an algebraic form of ordered pair notation gives <Vx, Vy> as the components of vector V.

2. Using i to represent the x axis, j to represent the y axis and k to represent the z axis. This gives a vector that has the form Vx i + Vy j + Vz k

Quadrant IV Example for converting angles as directed from the +x axis: mouse over to see how!

To add graphically using tail to tip method:

Start one vector where the preceeding vector stops. Here vector a (in blue) is added to vector b (in green).

The RESULTANT (vector sum) starts at the beginning of the first vector and terminates at the end of the last vector.

Reversing the direction of the vector makes it negative.

The red vector is vector a.

The blue vector is -a.

Note that vector a is the reverse of vector a. The vector sum is the red vector.

While it is best to have your vectors in rectangular form for addition, it is also necessary to be able to convert from rectangular form to polar form.

This is a two part process.

First, find the magnitude of the vector using the Pythagorean Theorem. MAGNITUDE of Vector V=

Next find the reference angle using trig relationships.
Tangent q = |(y component)| / |x component|

Note the absolute values! This gives your reference angle. Now you must find the correct quadrant! See to the right!

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What is a Vector?	A Vector is any quantity that requires both magnitude and direction to completely describe itself. Quantities in physics that only require magnitude are termed scalars. Scalars and Vectors		Examples of Vectors	Examples of Scalars
How to Name VECTORS!	A vector is usually named in one of two ways. *Using the POLAR FORM* way requires magnitude and then direction as in < R, q > (where R is magnitude and q is the angle from the +x axis).** Using the RECTANGULAR or COMPONENT method requires an x part and y part (2 dimensions) or for 3 dimensions an "i" (x axis), "j" (y axis), and "k" (z axis) written as <x,y,k> or 2 i - 4j + 7 k (an example for 3 dimensions).		Examples of Polar Vectors Polar form always states the magnitude first and then the direction. In physics, the direction is usually from the positive x axis and rotates counterclockwise following trig notation.	Examples of Component Vectors Two types of component notation are common. 1. Using an algebraic form of ordered pair notation gives <Vx, Vy> as the components of vector V. 2. Using i to represent the x axis, j to represent the y axis and k to represent the z axis. This gives a vector that has the form Vx i + Vy j + Vz k
How to Calculate Components from Polar Form!	We shall discuss only the x and y components of vectors. Discuss "direction cosines" for vectors in 3- space with your math teacher. To find the x component-- first, make sure your angle is given as directed from the + x axis! Then multiply the magnitude of the vector by the cosine of the angle so that Vector (x) = Vector Magnitude * cosine q. To find the y component-- first, make sure your angle is given as directed from the + x axis! Then multiply the magnitude of the vector by the sine of the angle so that Vector (y) = Vector Magnitude * sine q		Quadrant IV Example for converting angles as directed from the +x axis: mouse over to see how!	Quadrant II Example for converting angles :
How to Add Vectors	One of the most important uses of the component form is adding two or more vectors. To add vectors: Find all x components and all y components. Add all x components together keeing the sign of the sum. Add all y components together keeping the sign of the sum. The vector answer has both an "x" part and a "y" part generally written as <vx, vy> or vx i + vy j .		To add graphically using tail to tip method: Start one vector where the preceeding vector stops. Here vector a (in blue) is added to vector b (in green). The RESULTANT (vector sum) starts at the beginning of the first vector and terminates at the end of the last vector.	Add these two vectors: a = 4 m at 120 degrees b = 6 m at 210 degrees Graphically-mouse over to see sum! With Component Method
How to Subtract Vectors	Keep in mind that subtraction is just a form of addition. To obtain vector a- vector b, just add vector a to the negative of vector b. How to take the negative of a vector: Graphically, just reverse the direction. With components, reverse the signs of all parts. With polar form, add or subtract 180 degrees (it does not matter) to or from your angle.		*Reversing the direction of the vector makes it negative.* *The red vector is vector a.* *The blue vector is -a.*	*Note that vector a is the reverse of vector a. The vector sum is the red vector.*
Coverting to Polar Form	While it is best to have your vectors in rectangular form for addition, it is also necessary to be able to convert from rectangular form to polar form. This is a two part process. First, find the magnitude of the vector using the Pythagorean Theorem. MAGNITUDE of Vector V= Next find the reference angle using trig relationships. Tangent q = \|(y component)\| / \|x component\| Note the absolute values! This gives your reference angle. Now you must find the correct quadrant! See to the right!		Correct Direction? You should sketch or graph the vector. Use the POSITIVE x axis! IDK refers to identifying key! Quadrant 1-- IDK is positive x and positive y ! Your reference angle is your angle q. Quadrant 2-- IDK is negative x and positive y! Your angle is 180- q. Quadrant 3-- IDK is negative x and negative y! Your angle is 180 + q. Quadrant 4-- IDK is positive x and negative y! Your angle is 360 - q Did you observe that all angles are relative to the x-axis?	Look at the force vector (blue arrow). Name the force vector in component form.The unit is Newtons Now find the magnitude of the force vector in Newtons Last of all, find the direction of the vector
Multiplying Vectors	There are three types of vector multiplication: Vector by Scalar: changes length and changes direction only if the scalar is negative. Then the direction of the vector reverses 180 degrees. Vector "dot" Vector: a product that determines how much vectors point in the same direction. It is a scalar product! Vector "cross" Vector: a product that determines how perpendicular vectors are to each other. This product produces a vector answer!		More on "Dot" Products Equations:	More on "Cross" Products Equations: a X b = \|a\| \|b\| sine q with direction from right hand rule
			Vectorland	Where in the World?
Vectors, An Introduction		More INTRO to Vectors	Vector Basics	Take Mrs. Woolard's Test

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What is a Vector?

How to Name VECTORS!

Using the POLAR FORM way requires magnitude and then direction as in < R, q > (where R is magnitude and q is the angle from the +x axis).

Using the RECTANGULAR or COMPONENT method requires an x part and y part (2 dimensions) or for 3 dimensions an "i" (x axis), "j" (y axis), and "k" (z axis) written as <x,y,k> or 2 i - 4j + 7 k (an example for 3 dimensions).

Polar form always states the magnitude first and then the direction. In physics, the direction is usually from the positive x axis and rotates counterclockwise following trig notation.

Two types of component notation are common.

1. Using an algebraic form of ordered pair notation gives <Vx, Vy> as the components of vector V.

2. Using i to represent the x axis, j to represent the y axis and k to represent the z axis. This gives a vector that has the form Vx i + Vy j + Vz k

To find the x component-- first, make sure your angle is given as directed from the + x axis! Then multiply the magnitude of the vector by the cosine of the angle so that Vector (x) = Vector Magnitude * cosine q.

To find the y component-- first, make sure your angle is given as directed from the + x axis! Then multiply the magnitude of the vector by the sine of the angle so that Vector (y) = Vector Magnitude * sine q

Quadrant II Example for converting angles :

One of the most important uses of the component form is adding two or more vectors. To add vectors:

Find all x components and all y components.

Add all x components together keeing the sign of the sum.

Add all y components together keeping the sign of the sum.

The vector answer has both an "x" part and a "y" part generally written as <vx, vy> or vx i + vy j .

To add graphically using tail to tip method:

Start one vector where the preceeding vector stops. Here vector a (in blue) is added to vector b (in green).

The RESULTANT (vector sum) starts at the beginning of the first vector and terminates at the end of the last vector.

Add these two vectors:

a = 4 m at 120 degrees

b = 6 m at 210 degrees

Graphically-mouse over to see sum!

With Component Method

How to Subtract Vectors

Keep in mind that subtraction is just a form of addition. To obtain vector a- vector b, just add vector a to the negative of vector b.

How to take the negative of a vector:

Graphically, just reverse the direction.

With components, reverse the signs of all parts.

With polar form, add or subtract 180 degrees (it does not matter) to or from your angle.

Coverting to Polar Form

While it is best to have your vectors in rectangular form for addition, it is also necessary to be able to convert from rectangular form to polar form.

This is a two part process.

First, find the magnitude of the vector using the Pythagorean Theorem. MAGNITUDE of Vector V=

Next find the reference angle using trig relationships.

Note the absolute values! This gives your reference angle. Now you must find the correct quadrant! See to the right!

Correct Direction?

You should sketch or graph the vector. Use the POSITIVE x axis! IDK refers to identifying key!

Quadrant 1-- IDK is positive x and positive y ! Your reference angle is your angle q.

Quadrant 2-- IDK is negative x and positive y! Your angle is 180- q.

Quadrant 3-- IDK is negative x and negative y! Your angle is 180 + q.

Quadrant 4-- IDK is positive x and negative y! Your angle is 360 - q

Did you observe that all angles are relative to the x-axis?

Multiplying Vectors

There are three types of vector multiplication:

Vector by Scalar: changes length and changes direction only if the scalar is negative. Then the direction of the vector reverses 180 degrees.

Vector "dot" Vector: a product that determines how much vectors point in the same direction. It is a scalar product!

Vector "cross" Vector: a product that determines how perpendicular vectors are to each other. This product produces a vector answer!

More on "Dot" Products

Equations:

More on "Cross" Products

Equations:

a X b = |a| |b| sine q with direction from right hand rule

**Using the POLAR FORM way requires magnitude and then direction as in < R, q > (where R is magnitude and q is the angle from the +x axis).**