If d1 = 7 ihat - 2 jhat + 4 khat and d2 = -5 ihat + 2 jhat - khat, then what is (d1 + d2)·(d1 multiplied by 6 d2)?

Question

If d1 = 7 ihat - 2 jhat + 4 khat and d2 = -5 ihat + 2 jhat - khat, then what is (d1 + d2)·(d1 multiplied by 6 d2)?

Answer 1

To find the dot product of two vectors, we need to multiply their corresponding components and then sum them up.

Given d1 = 7 î - 2 ĵ + 4 k̂ and d2 = -5 î + 2 ĵ - k̂, let's first find the vector (d1 + d2) by adding the corresponding components:

(d1 + d2) = (7 î - 2 ĵ + 4 k̂) + (-5 î + 2 ĵ - k̂)
= (7 - 5) î + (-2 + 2) ĵ + (4 - 1) k̂
= 2 î + 0 ĵ + 3 k̂
= 2 î + 3 k̂

Next, we need to find the vector (d1 multiplied by 6 d2) by multiplying each component of d1 by 6 times the corresponding component of d2:

(d1 multiplied by 6 d2) = (6 * 7 î) + (6 * -2 ĵ) + (6 * 4 k̂) multiplied by (-5 î) + (6 * 2 ĵ) + (6 * -1 k̂)
= 42 î - 12 ĵ + 24 k̂ multiplied by -5 î + 12 ĵ - 6 k̂

Finally, we can find the dot product of the two vectors:

(d1 + d2) · (d1 multiplied by 6 d2) = (2 î + 3 k̂) · (42 î - 12 ĵ + 24 k̂) · (-5 î + 12 ĵ - 6 k̂)

To calculate the dot product, we multiply the corresponding components of the two vectors and then sum them up:

(2 î + 3 k̂) · (42 î - 12 ĵ + 24 k̂) = (2 * 42) + (3 * 0) + (24 * 3)
= 84 + 0 + 72
= 156

Therefore, (d1 + d2) · (d1 multiplied by 6 d2) = 156.