A vector component describes ...

Definition of Vector Component

A vector component is a projection of a vector onto the horizontal or vertical axis.

A vector directed northwest has components which are directed north and west. The northern component is the projection of the vector onto the north-south axis. The western component is the projection of the vector onto the east-west axis. If an airplane is flying northwest - with a velocity of say 200 km/hr at 120 degrees - then the plane has a northern and a western component of velocity. The northern component of velocity describes the effect of the plane's velocity in the northern direction. The western component of velocity describes the effect of the plane's velocity in the western direction. In Physics, components simply describe the effect of a vector in a given direction.

 What is a vector component?

 What is the significance of a vector component?

Consider the force vector shown in the diagram at the right. This force vector would have components which are directed ____ and ____.

Definition of Vector Component

A vector component is a projection of a vector onto the horizontal or vertical axis.

Don't be fooled by this question; there is nothing to be fooled by. A vector directed northwest has components which are directed north and west. The northern component is the projection of the vector onto the north-south axis. The western component is the projection of the vector onto the east-west axis. A vector directed southeast has components which are directed south and east. The southern component is the projection of the vector onto the north-south axis. The eastern component is the projection of the vector onto the east-west axis.

So to answer this question, simply look at the given vector and ask In which two directions is this vector pointing? The answer to your question is the answer you should pick.

Mathematically, a component is simply a projection of a vector onto an axis. To determine the components using a diagram, first draw the vector. Then sketch the x-y coordinate axes at the tail of the vector. From the arrowhead of the vector, sketch reference lines across to each of the coordinate axes such that the lines meet the axes with a perpendicular orientation to them. The x-component stretches along the x-axis from the tail of the vector to the location where the reference line meets the x-axis. The y-component stretches along the y-axis from the tail of the vector to the location where the reference line meets the y-axis.

If given a vector, how can one determine the direction of the components?

A vector is directed at a direction of 110 degrees (as measured counterclockwise from east). Such a vector would have two components. The ____ component would be greater than the ____ component.

(Note: Numbers are randomized numbers and likely different from the numbers listed here.)

Definition of Vector Component

A vector component is a projection of a vector onto the horizontal or vertical axis.

Counterclockwise from East Convention for Vector Direction

The direction of a vector is often expressed using the counterclockwise (CCW) convention. According to this convention, the direction of a vector is the number of degrees of rotation which the vector makes counterclockwise from East.

The components of a vector are often represented on a diagram by constructing a right triangle about the vector such that the vector is the hypotenuse of the right triangle. The components are then the legs of the right triangle. (You have likely seen such diagrams and might make an effort to sketch one now.) If the vector is a northwest vector, then it has components stretching west and north. The west and north components are simply the west and north legs of the triangle which has been created from the northwest vector.

Trigonometric functions can be used to determine the precise magnitude of the legs of these triangles. If the angle is the counterclockwise angle of rotation between the eastern axis and the vector, then the x-component can be calculated using the cosine function. The y-component can be calculated using the sine function.

A_x = A • cosine A_y = A • sine

where = counterclockwise angle rotation of vector A from east.

If given a vector, how can one determine the direction of the components?

How can trigonometric functions be used to determine the magnitude of the components of a vector?

What is meant by saying that a vector's direction is so many degrees counterclockwise from east?

A vector is shown in the diagram below. This vector has two components. The ____ component would be greater than the ____ component.

Definition of Vector Component

A vector component is a projection of a vector onto the horizontal or vertical axis.

The components of a vector are often represented on a diagram by constructing a right triangle about the vector such that the vector is the hypotenuse of the right triangle. The components are then the legs of the right triangle. (You have likely seen such diagrams and might make an effort to sketch one now.) If the vector is a northeast vector, then it has components stretching east and north. The east and north components are simply the east and north legs of the triangle which has been created from the northeast vector. If you wish to compare the magnitudes of the two components, simply compare the length of the legs on the triangle. The leg which is biggest (either the east-west leg or the north-south leg) is the component which is biggest.

The first step to successfully answering this question involves determining the direction of the two vector components - east or west and north or south. Simply look at the given vector and ask In which two directions is this vector pointing?

Once you have determined the direction of the two components, begin the comparison of their magnitudes. If a vector is directed northeast at 45 degrees into the first quadrant, then it is directed just as much east as it is north. But if the vector is rotated more than 45 degrees counterclockwise from East (that is more toward the north), then the northern component is bigger than the eastern component. And if the vector is rotated less than 45 degrees counterclockwise from East (that is more toward the east), then the eastern component is bigger than the northern component.

What is written here about a northeast component in the first quadrant can be written of any vector in any of the quadrants. The more a vector is rotated towards a given axis, the greater the component along the axis will be.

If given a vector, how can one determine the direction of the components?

How can trigonometric functions be used to determine the magnitude of the components of a vector?

Consider the diagram of a force vector shown below. The angle theta is expressed as the angle of the vector with respect to due north. As the angle theta INCREASES from the current value to 90 degrees (OR decreases from the current value to 0 degrees), the horizontal component of force ____ and the vertical component of force ____.

Definition of Vector Component

A vector component is a projection of a vector onto the horizontal or vertical axis.

Mathematically, a component is simply a projection of a vector onto an axis. To determine the components using a diagram, first draw the vector. Then sketch the x-y coordinate axes at the tail of the vector. From the arrowhead of the vector, sketch reference lines across to each of the coordinate axes such that the lines meet the axes with a perpendicular orientation to it. The x-component stretches along the x-axis from the tail of the vector to the location where the reference line meets the x-axis. The y-component stretches along the y-axis from the tail of the vector to the location where the reference line meets the y-axis.

A wise idea for this question is to pull out a page of scratch paper and sketch the vector in the direction as shown and label the angle . Then draw the projections of the vector onto the x- and y- axes (see Math Magic section above). Now visualize what would happen to the size of these components if the angle was increased towards 90 degrees (or decreased towards 0 degrees). Would the horizontal leg of the triangle get bigger or smaller? Would the vertical leg of the triangle get bigger or smaller?

On a sheet of scratch paper, draw a large x-y coordinate axes. Use a pencil or a pen as a vector and place one end (the tail end) at the origin. Point the other end of the pencil or pen in the direction of the vector. Visualize the components of the vector - maybe even sketch them on the paper. Now rotate the vector (pencil or pen) in the indicated direction and ask what happens to the magnitude of the horizontal and vertical components. You might notice that as the vector gets closer and closer to the x-axis, the size of the vertical component decreases and the size of the horizontal component increases. The opposite is true of a vector which gets closer and closer to the y-axis.

How can one determine the magnitude of the components of a vector?

Suppose that a force with a magnitude of 42.6 N is exerted at an angle of 36 degrees with the horizontal. This force would be the same as having two forces which are exerted at ____ and ____.

(Note: Numbers are randomized numbers and likely different from the numbers listed here.)

Definition of Vector Component

A vector component is a projection of a vector onto the horizontal or vertical axis.

The components of a vector are often represented on a diagram by constructing a right triangle about the vector such that the vector is the hypotenuse of the right triangle. The components are then the legs of the right triangle. (You have likely seen such diagrams and might make an effort to sketch one now.) If the vector is a northwest vector, then it has components stretching west and north. The west and north components are simply the west and north legs of the triangle which has been created from the northwest vector.

Trigonometric functions can be used to determine the precise magnitude of the legs of these triangles. If the angle is the counterclockwise angle of rotation from the eastern axis and the vector, then the x-component can be calculated using the cosine function; the y-component can be calculated using the sine function.

A_x = A • cosine A_y = A • sine

where = counterclockwise angle rotation of vector A from the east.

How can trigonometric functions be used to determine the magnitude of the components of a vector?

Suppose that a force with a magnitude of 54.6 N is exerted at a direction of 160 degrees (expressed as a counterclockwise angle of rotation from due east). This force would be the same as having two forces which are exerted at ____ and ____.

(Note: Numbers are randomized numbers and likely different from the numbers listed here.)

Definition of Vector Component

A vector component is a projection of a vector onto the horizontal or vertical axis.

The components of a vector are often represented on a diagram by constructing a right triangle about the vector such that the vector is the hypotenuse of the right triangle. The components are then the legs of the right triangle. (You have likely seen such diagrams and might make an effort to sketch one now.) If the vector is a northwest vector, then it has components stretching west and north. The west and north components are simply the west and north legs of the triangle which has been created from the northwest vector.

Trigonometric functions can be used to determine the precise magnitude of the legs of these triangles. If the angle is the counterclockwise angle of rotation from the eastern axis and the vector, then the x-component can be calculated using the cosine function. The y-component can be calculated using the sine function.

A_x = A • cosine A_y = A • sine

where = counterclockwise angle rotation of vector A from the east.

How can trigonometric functions be used to determine the magnitude of the components of a vector?