If a is a unit vector Ibl = 5 and la + b1 = square root17. determine (-4a - 3b) • (b + 7a)
|a| = 1
|b| = 5
|a+b| = √17
(-4a - 3b) • (7a + b) = -28a•a - 25a•b - 3b•b
= -28 - 15 - 25a•b
= -43 - 25a•b
See what you can do with that, and the definition of a•b
To find the dot product of the vectors (-4a - 3b) and (b + 7a), we need to compute the scalar product of their corresponding components.
Given:
Ibl = 5 (magnitude of b)
la + b1 = square root(17)
To solve for a and b, we can form two equations using the above information:
1. |b| = 5
2. |a + b| = sqrt(17)
To find a, let's square both sides of equation 2:
(a + b) • (a + b) = (sqrt(17))^2
(a • a) + 2(a • b) + (b • b) = 17
1 + 2(a • b) + 25 = 17 [since a • a = 1 and b • b = 5^2]
2(a • b) = -9
(a • b) = -4.5 [Dividing both sides by 2]
Now, we can calculate (-4a - 3b) • (b + 7a):
(-4a - 3b) • (b + 7a)
= (-4a • b) + (-4a • 7a) + (-3b • b) + (-3b • 7a) [Distribute the dot product]
= -4(a • b) - 28(a • a) - 3(b • b) - 21(b • a) [Simplify by substituting values]
= -4(-4.5) - 28(1) - 3(25) - 21(-4.5) [Substitute (a • b) = -4.5, (a • a) = 1, (b • b) = 5^2, (b • a) = (a • b)]
= 18 + (-28) - 75 + 94.5
= 9.5
Therefore, (-4a - 3b) • (b + 7a) equals 9.5.