Given \( v=10 i+6 j \) and \( w=-i+1 \). a. Find prosw b. Decompose \( v \) into two vectors \( v_{1} \) and \( v_{2} \), where \( v_{1} \) is parallel to \( w \) and \( v_{2} \) is orthogonal to \( w \) a. projw \( v=\square \) (Type your answer in terms of \( i \) and \( j \).) b. \( v_{1}=\square \) (Type your answer in terms of \( i \) and \( j \).) \( v_{2}=\square \) (Type your answer in terms of \( i \) and \( j \).)

To solve this problem, we need to approach parts (a) and (b) step-by-step. ### a. Find the projection of \( v \) onto \( w \) The projection of vector \( v \) onto vector \( w \) is given by the formula: \[ \text{proj}_w v = \frac{v \cdot w}{w \cdot w} w \] First, we calculate the dot products: - \( v \cdot w = (10i + 6j) \cdot (-i + 1j) = 10(-1) + 6(1) = -10 + 6 = -4 \) - \( w \cdot w = (-i + j) \cdot (-i + j) = (-1)(-1) + (1)(1) = 1 + 1 = 2 \) Now substituting back into the projection formula: \[ \text{proj}_w v = \frac{-4}{2} w = -2w = -2(-i + j) = 2i - 2j \] So, the answer for part (a) is: \[ \text{proj}_w v = 2i - 2j \] ### b. Decomposing \( v \) into \( v_1 \) and \( v_2 \) From the results above, we found \( v_1 \) as the projection of \( v \) onto \( w \): \[ v_1 = 2i - 2j \] To find \( v_2 \), we simply subtract \( v_1 \) from \( v \): \[ v_2 = v - v_1 = (10i + 6j) - (2i - 2j) = (10i - 2i) + (6j + 2j) = 8i + 8j \] So, the answers for part (b) are: \[ v_1 = 2i - 2j \] \[ v_2 = 8i + 8j \] ### Final Answers a. \(\text{proj}_w v = 2i - 2j\) b. \(v_1 = 2i - 2j\), \(v_2 = 8i + 8j\)