Questions: Find 2/3 u, 3 v, v-u, and 2 u+5 v u=⟨7,9⟩, v=(6,-6⟩ (a) 2/3 u (b) 3 v (c) v-u (d) 2 u+5 v

Find 2/3 u, 3 v, v-u, and 2 u+5 v
u=⟨7,9⟩, v=(6,-6⟩
(a) 2/3 u 
(b) 3 v 
(c) v-u 
(d) 2 u+5 v
Transcript text: Find $\frac{2}{3} \mathbf{u}, 3 \mathbf{v}, \mathbf{v}-\mathbf{u}$, and $2 \mathbf{u}+5 \mathbf{v}$ \[ \mathbf{u}=\langle 7,9\rangle, \mathbf{v}=(6,-6\rangle \] (a) $\frac{2}{3} \mathbf{u}$ $\square$ (b) $3 v$ $\square$ (c) $\mathbf{v}-\mathbf{u}$ $\square$ (d) $2 \mathbf{u}+5 \mathbf{v}$ $\square$
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Solution

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Solution Steps

To solve the given vector operations, we will:

  1. Multiply vector u\mathbf{u} by 23\frac{2}{3}.
  2. Multiply vector v\mathbf{v} by 3.
  3. Subtract vector u\mathbf{u} from vector v\mathbf{v}.
  4. Add 2 times vector u\mathbf{u} to 5 times vector v\mathbf{v}.
Step 1: Calculate 23u\frac{2}{3} \mathbf{u}

To find 23u\frac{2}{3} \mathbf{u}, we multiply each component of u\mathbf{u} by 23\frac{2}{3}: u=7,9 \mathbf{u} = \langle 7, 9 \rangle 23u=237,239=4.6667,6 \frac{2}{3} \mathbf{u} = \left\langle \frac{2}{3} \cdot 7, \frac{2}{3} \cdot 9 \right\rangle = \left\langle 4.6667, 6 \right\rangle

Step 2: Calculate 3v3 \mathbf{v}

To find 3v3 \mathbf{v}, we multiply each component of v\mathbf{v} by 3: v=6,6 \mathbf{v} = \langle 6, -6 \rangle 3v=36,36=18,18 3 \mathbf{v} = \left\langle 3 \cdot 6, 3 \cdot -6 \right\rangle = \left\langle 18, -18 \right\rangle

Step 3: Calculate vu\mathbf{v} - \mathbf{u}

To find vu\mathbf{v} - \mathbf{u}, we subtract each component of u\mathbf{u} from the corresponding component of v\mathbf{v}: vu=67,69=1,15 \mathbf{v} - \mathbf{u} = \left\langle 6 - 7, -6 - 9 \right\rangle = \left\langle -1, -15 \right\rangle

Step 4: Calculate 2u+5v2 \mathbf{u} + 5 \mathbf{v}

To find 2u+5v2 \mathbf{u} + 5 \mathbf{v}, we first multiply each component of u\mathbf{u} by 2 and each component of v\mathbf{v} by 5, then add the corresponding components: 2u=27,29=14,18 2 \mathbf{u} = \left\langle 2 \cdot 7, 2 \cdot 9 \right\rangle = \left\langle 14, 18 \right\rangle 5v=56,56=30,30 5 \mathbf{v} = \left\langle 5 \cdot 6, 5 \cdot -6 \right\rangle = \left\langle 30, -30 \right\rangle 2u+5v=14+30,18+30=44,12 2 \mathbf{u} + 5 \mathbf{v} = \left\langle 14 + 30, 18 + -30 \right\rangle = \left\langle 44, -12 \right\rangle

Final Answer

(a) 23u=4.6667,6\frac{2}{3} \mathbf{u} = \boxed{\left\langle 4.6667, 6 \right\rangle}

(b) 3v=18,183 \mathbf{v} = \boxed{\left\langle 18, -18 \right\rangle}

(c) vu=1,15\mathbf{v} - \mathbf{u} = \boxed{\left\langle -1, -15 \right\rangle}

(d) 2u+5v=44,122 \mathbf{u} + 5 \mathbf{v} = \boxed{\left\langle 44, -12 \right\rangle}

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