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