Understanding Perpendicular Vector Components

In summary, a vector with nonzero magnitude will always have a component of zero length in the direction perpendicular to the vector. This can be visualized by imagining a vector rising perpendicularly from a line and observing its projection onto the line. Similarly, in the Cartesian plane, the amount of "y" along the direction of "x" is determined by the perpendicular vector. This can also be demonstrated by drawing vectors A and B with their starting ends together and observing that vector C, which is the sum of A and B, will have the same length as A when B is small enough.
  • #1
fightboy
25
0
"Is it possible for a vector that has nonzero magnitude to have a component in some direction that is equal to zero?"
The answer key said that any vector that has a nonzero magnitude will always have a component of zero length in the direction perpendicular to the vector.

I'm having trouble visualizing this. Why will the vector always have a component of zero length?
If anyone could break this down for me it would be much appreciated!
 
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  • #2
Imagine a vector rising perpendicularly from a line (any given line). What's the projection of that vector onto the line?

An obvious analogy is the x-y axes of the Cartesian plane. How much "y" lies along the direction of "x"?
 
  • #3
Draw a vector A on paper. Now, draw a vector B perpendicular to the A vector, both A and B have their startying ends together. Now draw a vector C from the tip of A to the tip of B. You obviously have C = A + B. How small does B have to be so that A = C?
 

FAQ: Understanding Perpendicular Vector Components

1. What are vector components?

Vector components refer to the individual parts or directions that make up a vector quantity. They are often represented by the x, y, and z axes in three-dimensional space.

2. How do you find the components of a vector?

To find the components of a vector, you can use trigonometric functions such as sine and cosine. For example, the x component of a vector can be found by multiplying the magnitude of the vector by the cosine of the angle between the vector and the x-axis.

3. What is the difference between vector components and scalar components?

Vector components are directional and have both magnitude and direction, while scalar components are only numerical values with no direction. Vector components are used to represent quantities such as force, velocity, and acceleration, while scalar components are used for quantities like mass, temperature, and time.

4. Can vector components be negative?

Yes, vector components can be negative. This occurs when the vector is pointing in the negative direction of a given axis. For example, a vector with an x component of -5 would be pointing to the left on the x-axis.

5. How are vector components used in physics?

Vector components are used in physics to break down complex vector quantities into simpler components that are easier to work with. They are also used to analyze motion, forces, and other physical phenomena in different directions and dimensions.

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