Given the vectors a = (1, 3, 4) and b = (4, 5, -4), what is the angle between the two given vectors to the nearest degree?

Answer 1

Calculate the cross product using the matrix determinant rule. That will tell you the sine of the angle.

The magnitudes of the two vectors are
sqrt(1+9+16) = 5.099 and sqrt(16 + 25 + 16) = 7.550

The cross product is
|i j k |
|1 3 4 |
|4 5 -4 |
= -24 i + 20 j -7 k
The magnitude of the cross product is
sqrt (576 + 400 + 49) = 32.015

sin theta = 32.015 /[(5.099)(7.550)]
= 0.8316
theta = 56.3 degrees

Answer 2

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