Question

Define two vector functions: → r(t) = 10 sin(t) → i + 7 cos(t) → j + 8t → k → u(t) = 7 sin(t) → i + 10 cos(t) → j + (t^3 - 8) → k What is the position vector → r at t = π/4 for each function? a) → r = (10√2 → i + 7√2 → j + 2√2 → k), → u = (7√2 → i + 10√2 → j + (√2)^3 → k) b) → r = (10√2 → i + 7√2 → j + 8√2 → k), → u = (7√2 → i + 10√2 → j + (2√2)^3 → k) c) → r = (10√2 → i + 7√2 → j + 8√2 → k), → u = (7√2 → i + 10√2 → j + (√2)^3 → k)

Answer 1

First of all, we need to plug t = π/4 into our vector functions → r(t) and → u(t). Let's start with → r(t).

To find the → i, → j and → k components of → r, we have to substitute t = π/4 into all three component functions:

We get for → i component: 10 sin(π/4) which results in 7.07106781 when calculated.
For → j component: We have 7 cos(π/4), upon calculating this we get 4.94974747.
For → k component: When we substitute t = π/4 into 8t we get 6.28318531.

So, the position vector → r at t = π/4 becomes → r = 7.07106781 → i + 4.94974747 → j + 6.28318531 → k.

Next, we substitute t = π/4 into our → u(t) function:

The → i component becomes 7 sin(π/4), equal to 4.94974747.
The → j component is 10 cos(π/4), which results in 7.07106781.
The → k component is given by (t^3 - 8) and when we substitute t = π/4 into it; we get -7.51552693.

So, the position vector → u at t = π/4 is → u = 4.94974747 → i + 7.07106781 → j - 7.51552693 → k.

From our options in the question, the result we got matches with none of them. Thus, we inferred that there might be a miscalculation in the options provided. Comparing our calculated values with the options, we see that the closest option is C. However, even though option C looks closest to our calculated value, it is not the exact match.