Download Interlude 1: Exercise 4

Interlude 1: Exercise 4
Explanation
This exercise is a also straight algebraic manipulation, providing practice in calculating with vectors.
Hint
Recall the definition of the magnitude of a vector, the definition of the dot product, and the use of trigonometry in calculating the angle
between vectors.
Answer
The magnitude of a vector can be calculated using Eq. (3).
£A§ =
®
and
£B§ =
®
A x 2 + A y 2 + Az 2 =
B x 2 + B y 2 + Bz 2 =
22 + H-3L2 + 12 =
H-4L2 + H-3L2 + 22 =
4+9+1 =
16 + 9 + 4 =
14 » 3.74166
29 » 5.38516.
2
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The dot product can be calculated using
A × B = Ax Bx + A y B y + Az Bz = H2L H-4L + H-3L H-3L + H1L H2L = -8 + 9 + 2 = 3.
® ®
The angle between the vectors can be calculated by using the other definition of the dot product,
A × B = £A§ £B§ cos Θ,
® ®
we can solve this for cos Θ,
®
®
® ®
A×B
cos Θ =
£A§ £B§
®
®
3
=
H3.74166L H5.38516L
then we take the arccos Θ, to find that Θ » 1.42135 radians, or about 81.4374°.
» 0.148888