Question
These are the component forms of vectors \( \vec{p} \) and \( \vec{q} \) : \( \vec{p}=(5,0) \) \( \vec{q}=(4,-3) \) Colten and Martha tried to find the difference \( \vec{p}-\vec{q} \) of the vectors.
Ask by Cross Rowe. in Iceland
Mar 11,2025
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Answer
The difference \( \vec{p} - \vec{q} \) is \( (1, 3) \).
Solution
To find the difference of the vectors \( \vec{p} \) and \( \vec{q} \), we can use the component forms of the vectors. The difference of two vectors is calculated by subtracting their corresponding components.
Given:
- \( \vec{p} = (5, 0) \)
- \( \vec{q} = (4, -3) \)
The difference \( \vec{p} - \vec{q} \) can be calculated as follows:
\[
\vec{p} - \vec{q} = (p_1 - q_1, p_2 - q_2)
\]
Where:
- \( p_1 = 5 \), \( p_2 = 0 \)
- \( q_1 = 4 \), \( q_2 = -3 \)
Now, substituting the values:
1. Calculate the first component:
\[
p_1 - q_1 = 5 - 4 = 1
\]
2. Calculate the second component:
\[
p_2 - q_2 = 0 - (-3) = 0 + 3 = 3
\]
Putting it all together, we have:
\[
\vec{p} - \vec{q} = (1, 3)
\]
Thus, the difference of the vectors \( \vec{p} \) and \( \vec{q} \) is \( (1, 3) \).
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Simplify this solution Bonus Knowledge
To find the difference \( \vec{p} - \vec{q} \), you simply subtract the corresponding components of the vectors. So, you take the \( x \)-components: \( 5 - 4 = 1 \) and the \( y \)-components: \( 0 - (-3) = 0 + 3 = 3 \). Therefore, the result of the operation is \( \vec{p} - \vec{q} = (1, 3) \). In vector mathematics, visualizing these vectors on a coordinate plane can be super helpful! Imagine \( \vec{p} \) pointing directly to the right along the x-axis while \( \vec{q} \) goes down and to the left. Drawing them out not only makes subtraction manageable but also reveals how they relate in terms of direction and magnitude—perfect for getting a geometric feel for the math!
