Let $$ \mathbf{u}=\langle 5,-2\rangle, \mathbf{v}=\langle-2,8\rangle $$, and $$ \mathbf{w}=\langle 0,-6\rangle $$. Express $$ 7 \mathbf{u}-2 \mathbf{v}+\mathbf{w} $$ in the form $$ \langle\mathrm{a}, \mathrm{b}\rangle $$. $$ 7 \mathbf{u}-2 \mathbf{v}+\mathbf{w}=\langle\square $$ (Simplify your answers.)

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UpStudy AI · Accepted Answer

To express $$ 7 \mathbf{u} - 2 \mathbf{v} + \mathbf{w} $$ in the form $$ \langle a, b \rangle $$, follow these steps:

1. **Multiply each vector by its scalar:**
   $$
   7\mathbf{u} = 7\langle 5, -2 \rangle = \langle 35, -14 \rangle
   $$
   $$
   -2\mathbf{v} = -2\langle -2, 8 \rangle = \langle 4, -16 \rangle
   $$
   $$
   \mathbf{w} = \langle 0, -6 \rangle
   $$

2. **Add the resulting vectors component-wise:**
   $$
   7\mathbf{u} - 2\mathbf{v} + \mathbf{w} = \langle 35, -14 \rangle + \langle 4, -16 \rangle + \langle 0, -6 \rangle
   $$
   $$
   = \langle 35 + 4 + 0, -14 - 16 - 6 \rangle
   $$
   $$
   = \langle 39, -36 \rangle
   $$

**Final Answer:**
$$
7 \mathbf{u} - 2 \mathbf{v} + \mathbf{w} = \langle 39, -36 \rangle
$$
$$ 7 \mathbf{u} - 2 \mathbf{v} + \mathbf{w} = \langle 39, -36 \rangle $$

Let , and . Express in the form .

(Simplify your answers.)

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