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

Solution Steps

To solve the given vector operations, we will:

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

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

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

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

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

Final Answer