Let \( \vec{a}=\langle-5,4,2\rangle \) and \( \vec{b}=\langle-4,-2,5\rangle \) Find the component of \( \vec{b} \) onto \( \vec{a} \).

To find the component of vector \( \vec{b} \) onto vector \( \vec{a} \), we can use the formula: \[ \text{comp}_{\vec{a}} \vec{b} = \frac{\vec{a} \cdot \vec{b}}{|\vec{a}|^2} \vec{a} \] First, we need to compute the dot product \( \vec{a} \cdot \vec{b} \): \[ \vec{a} \cdot \vec{b} = (-5)(-4) + (4)(-2) + (2)(5) = 20 - 8 + 10 = 22 \] Next, we find \( |\vec{a}|^2 \): \[ |\vec{a}|^2 = (-5)^2 + 4^2 + 2^2 = 25 + 16 + 4 = 45 \] Now we can calculate the component: \[ \text{comp}_{\vec{a}} \vec{b} = \frac{22}{45} \vec{a} \] Now substituting \( \vec{a} \): \[ \text{comp}_{\vec{a}} \vec{b} = \frac{22}{45} \langle -5, 4, 2 \rangle = \left\langle \frac{22}{45} \cdot -5, \frac{22}{45} \cdot 4, \frac{22}{45} \cdot 2 \right\rangle \] Calculating each component: \[ = \left\langle -\frac{110}{45}, \frac{88}{45}, \frac{44}{45} \right\rangle \] Thus, the component of \( \vec{b} \) onto \( \vec{a} \) is \[ \text{comp}_{\vec{a}} \vec{b} = \left\langle -\frac{110}{45}, \frac{88}{45}, \frac{44}{45} \right\rangle \]