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Short answer📘 N.VM.B.5🧠 DOK 3

Question:

Given vectors u=⟨3,−2⟩ \mathbf{u} = \langle 3, -2 \rangle and v=⟨−1,4⟩ \mathbf{v} = \langle -1, 4 \rangle , demonstrate how vector subtraction can be understood as adding the additive inverse. Calculate u−v \mathbf{u} - \mathbf{v} and explain each step of your solution.

✅ Answer:
To find \( \mathbf{u} - \mathbf{v} \), we take the vector \( \mathbf{u} = \langle 3, -2 \rangle \) and add the additive inverse of \( \mathbf{v} = \langle -1, 4 \rangle \). The additive inverse of \( \mathbf{v} \) is \( -\mathbf{v} = \langle 1, -4 \rangle \). Thus, \( \mathbf{u} - \mathbf{v} = \mathbf{u} + (-\mathbf{v}) = \langle 3, -2 \rangle + \langle 1, -4 \rangle = \langle 3+1, -2-4 \rangle = \langle 4, -6 \rangle \).

💡 Explanation:

  • Vector subtraction u−v \mathbf{u} - \mathbf{v} is the same as adding u \mathbf{u} to the additive inverse of v \mathbf{v} .

  • Here, the additive inverse of v \mathbf{v} is −v=⟨1,−4⟩ -\mathbf{v} = \langle 1, -4 \rangle because negating each component of v \mathbf{v} results in its opposite.

  • By adding u \mathbf{u} and −v -\mathbf{v} , we perform component-wise addition: 3+1=4 3 + 1 = 4 and −2−4=−6 -2 - 4 = -6 .

  • Therefore, u−v=⟨4,−6⟩ \mathbf{u} - \mathbf{v} = \langle 4, -6 \rangle .

Standard: N.VM.B.5 | DOK: 3

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