Certainly! Let's consider the concept of a vector in a 2D plane and examine the situation where one of its components is non-zero and the other is zero.
A vector can be represented in a 2D coordinate system by two components: [tex]\( \mathbf{v} = (a, b) \)[/tex], where [tex]\( a \)[/tex] and [tex]\( b \)[/tex] are the magnitudes of the vector in the x and y directions, respectively.
Example:
Let's take the vector [tex]\( \mathbf{v} = (5, 0) \)[/tex], where:
- The x-component ([tex]\( a \)[/tex]) is 5,
- The y-component ([tex]\( b \)[/tex]) is 0.
The magnitude of a vector is given by the formula:
[tex]\[
|\mathbf{v}| = \sqrt{a^2 + b^2}
\][/tex]
For our example:
[tex]\[
|\mathbf{v}| = \sqrt{5^2 + 0^2}
\][/tex]
Breaking it down step-by-step:
1. Square the x-component: [tex]\( 5^2 = 25 \)[/tex],
2. Square the y-component: [tex]\( 0^2 = 0 \)[/tex],
3. Add the results: [tex]\( 25 + 0 = 25 \)[/tex],
4. Take the square root of the sum: [tex]\( \sqrt{25} = 5 \)[/tex].
Thus, the magnitude of the vector [tex]\( (5, 0) \)[/tex] is 5.
So, for the given vector [tex]\( (5, 0) \)[/tex]:
- The x-component is non-zero (5),
- The y-component is zero,
- The magnitude of the vector is 5.
This example shows that if one of the components of a vector is non-zero (in this case, the x-component), the magnitude of the other component (the y-component) can indeed be zero.
