Questions: Pretest: Complex Numbers, Matrices, and Vectors Drag the tiles to the correct boxes to complete the pairs. Not all tiles will be used. Given: u=⟨-4,-1⟩, v=⟨0,-3⟩, w=⟨1,-1⟩, and z=⟨4,2⟩. Based on the components of the given vectors, match the magnitude of each vector subtraction with its resulting value. 8.54 1.83 3.61

Solution Steps

Given $\mathbf{u} = \langle -4, -1 \rangle$ and $\mathbf{z} = \langle 4, 2 \rangle$, $\mathbf{u} - \mathbf{z} = \langle -4 - 4, -1 - 2 \rangle = \langle -8, -3 \rangle$. $||\mathbf{u} - \mathbf{z}|| = \sqrt{(-8)^2 + (-3)^2} = \sqrt{64 + 9} = \sqrt{73} \approx 8.54$.

Given $\mathbf{v} = \langle 0, -3 \rangle$ and $\mathbf{w} = \langle 1, -1 \rangle$, $\mathbf{v} - \mathbf{w} = \langle 0 - 1, -3 - (-1) \rangle = \langle -1, -2 \rangle$. $||\mathbf{v} - \mathbf{w}|| = \sqrt{(-1)^2 + (-2)^2} = \sqrt{1 + 4} = \sqrt{5} \approx 2.24$.

Given $\mathbf{w} = \langle 1, -1 \rangle$ and $\mathbf{u} = \langle -4, -1 \rangle$, $\mathbf{w} - \mathbf{u} = \langle 1 - (-4), -1 - (-1) \rangle = \langle 5, 0 \rangle$. $||\mathbf{w} - \mathbf{u}|| = \sqrt{5^2 + 0^2} = \sqrt{25} = 5$.

Given $\mathbf{u} = \langle -4, -1 \rangle$ and $\mathbf{w} = \langle 1, -1 \rangle$, $-\mathbf{u} - \mathbf{w} = \langle -(-4) - 1, -(-1) - (-1) \rangle = \langle 4-1, 1+1 \rangle = \langle 3, 2\rangle$ $||-\mathbf{u} - \mathbf{w}|| = \sqrt{3^2+2^2} = \sqrt{9+4} = \sqrt{13} \approx 3.61$

Final Answer