Given vectors u= (t, 2t, 3) and v=(2t, 5t, -1). For what values of t
a) are u and v parallel?
b) are u and v perpendicular?
to be parallel u = kv
or in our case: 2t/t = 5t/2t = -1/3
or
2/1 = 5/2 = -1/3 which is a contradiction, so
no value of t makes them parallel
to be perp. u•v = 0
2t^2 + 10t^2 - 3 = 0
12t^2 = 3
t^2 = 1/4
t = ± 1/2
To determine if two vectors are parallel, we need to check if they are scalar multiples of each other. If two vectors are scalar multiples, then they have the same direction, although the magnitude may differ.
a) To check if u and v are parallel, we will set up the proportion by equating the corresponding components of the vectors:
$$\frac{t}{2t} = \frac{2t}{5t} = \frac{3}{-1}$$
Simplifying the equations gives:
$$\frac{1}{2} = \frac{2}{5} = -3$$
Since the ratios are not equal, there are no values of t that make u and v parallel.
b) To check if u and v are perpendicular, we can use the dot product. Two vectors are perpendicular if their dot product is zero.
The dot product of u and v can be found by multiplying the corresponding components and summing them up:
$$u \cdot v = (t \cdot 2t) + (2t \cdot 5t) + (3 \cdot -1) = 2t^2 + 10t^2 - 3$$
To find the values of t for which u and v are perpendicular, we set the dot product equal to zero:
$$2t^2 + 10t^2 - 3 = 0$$
Combining the terms:
$$12t^2 - 3 = 0$$
Solving this quadratic equation for t:
$$12t^2 = 3$$
$$t^2 = \frac{3}{12}$$
$$t^2 = \frac{1}{4}$$
Taking the square root of both sides:
$$t = \pm \frac{1}{2}$$
Therefore, the values of t for which u and v are perpendicular are t = -1/2 and t = 1/2.