(i) To find a such that v and u are orthogonal, we need to find the dot product of v and u and set it equal to zero:
v • u = ai * 4 + aj * 3 = 0
a * 4 + 3 = 0
4a = -3
a = -3/4
Therefore, when a = -3/4, v and u are orthogonal.
(ii) To find a such that v and u are parallel, we need to find the scalar multiple that makes one vector a multiple of the other:
v = ai + aj = u
Therefore, when a = 1, v and u are parallel.
(iii) To find a such that the angle between v and u is 30°, we can use the dot product formula:
v • u = |v||u|cosθ
Substitute the given values:
(ai + aj) • (4i + 3j) = √(a^2 + 1) * √(4^2 + 3^2) * cos(30°)
(4a + 3) = √(a^2 + 1) * 5 * √(3)/2
4a + 3 = 5√(a^2 + 1) * √(3)/2
8a + 6 = 15√(a^2 + 1)
64a^2 + 96a + 36 = 225a^2 + 225
161a^2 - 96a - 189 = 0
Solving this quadratic equation will give us the value of a that makes the angle between v and u 30°.
Let v ai j = + and 4 3 u i j = + , find (a) so that
(i) v and u are orthogonal.
(ii) v and u are parallel.
(ii) The angle between v and u is 30o
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