vector-valued functions?

Question

find the speed of an object having the position vector:
r(t) = sin(t) i +sqrt(2) cos(t) j + sin(t) k

Answer 1

You can differentiate each of the component vector with respect to t to get the velocity components. The speed is then the vector sum (square root of the sum of the squares).

x(t) = cos(t)
y(t) = -√2sin(t)
z(t) = cos(t)

v(t) = √[cos^2(t) + 2sin^2(t) + cos^2(t)]

since sin^2 + cos^2 = 1 this can be expressed as

v(t) = √[cos^2(t) + sin^2(t) + cos^2(t) + sin^2(t)]

v(t) = √2

Answer 2

v(t) = r'(t) = cos(t) i - sqrt2 cos(t) j + cos(t) k