Given \( v=3 i-4 j \) and \( w=-9 i+8 j \), find the following. \( \begin{array}{ll}\text { a) } v \cdot w & \text { b) } v \cdot v \\ \text { a) } v \cdot w=\square \\ \text { b) } v \cdot v=\square\end{array} \)

To solve the problems, we will use the formula for the dot product of two vectors. For two vectors \( v = a_1 i + b_1 j \) and \( w = a_2 i + b_2 j \), the dot product is given by: \[ v \cdot w = a_1 a_2 + b_1 b_2 \] **a)** For \( v \cdot w \): - \( v = 3 i - 4 j \) (where \( a_1 = 3 \) and \( b_1 = -4 \)) - \( w = -9 i + 8 j \) (where \( a_2 = -9 \) and \( b_2 = 8 \)) Now calculate: \[ v \cdot w = (3)(-9) + (-4)(8) = -27 - 32 = -59 \] So, \( v \cdot w = -59 \). **b)** For \( v \cdot v \): - Here, we will use the same vector \( v \): \[ v \cdot v = (3)(3) + (-4)(-4) = 9 + 16 = 25 \] So, \( v \cdot v = 25 \). Combining our results: \[ \begin{array}{ll} \text { a) } v \cdot w & = -59 \\ \text { b) } v \cdot v & = 25 \end{array} \]