Find two unit vectors that make an angle of 60° with v = 8, 6
Solution:
It is given that
Angle = 60°
v = 8, 6
Consider unit vector a = (x, y)
x2 + y2 = 1 …. (1)
Now apply the dot product of unit vector with v
a.v = |a||v|cos Ø
a.v = √ (x2 + y2) √ (82 + 62) cos 60º
8x + 6y = 1 × 10 × ½
So we get
8x + 6y = 5 …. (2)
y = (5 - 8x) / 6
Substituting the value of y in equation (1)
x2 + y2 = 1
x2 + [(5 - 8x)/6]2 = 1
x2 + 25/36 + 16x2/9 - 20x/9 = 1
25x2/9 - 20x/9 = 11/36
So we get
4 × (25x2 - 20x) = 11
100x2 - 80x = 11
100x2 - 80x - 11 = 0
Now solve the quadratic
x = [-b +-sqrt(b2 - 4ac)]/2a
x= { 2/5 ± √5/10}
Substituting the value of x in equation (2)
y = 3/10 ± 2√5/15
Therefore, the two unit vectors are (2/5 +√5/10, 3/10 + 2√5/15 ) and (2/5 -√5/10, 3/10 - 2√5/15).
Find two unit vectors that make an angle of 60° with v = 8, 6
Summary:
The two unit vectors that make an angle of 60° with v = 8, 6 are (2/5 +√5/10, 3/10 + 2√5/15 ) and (2/5 -√5/10, 3/10 - 2√5/15).
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