Evaluate (if possible) the vector-valued function at each...

Evaluate (if possible) the vector-valued function at each given value of $t$.
$\mathbf{r}(t)=\cos t \mathbf{i}+2 \sin t \mathbf{j}$
(a) $\mathbf{r}(0)$
(b) $\mathbf{r}(\pi / 4)$
(c) $\mathbf{r}(\theta-\pi)$
(d) $\mathbf{r}(\pi / 6+\Delta t)-\mathbf{r}(\pi / 6)$

Key Concepts

Trigonometric Functions

Trigonometric functions such as sine and cosine often appear in vector-valued functions to describe periodic behavior or circular motion. Understanding their properties and values at various angles is crucial for accurately computing the resulting vectors.

Vector Operations

Vector operations include addition, subtraction, and scalar multiplication, which are performed component-wise. In the context of vector-valued functions, operations like finding the difference between vectors correspond to computing changes or increments of the function with respect to the parameter.

Vector-Valued Functions

These functions assign a vector to each value of a parameter, usually represented as a combination of scalar functions multiplied by unit vectors. They are essential in describing curves and motions in space by showing how each coordinate changes with respect to the parameter.

Parameter Substitution

Parameter substitution involves directly replacing the variable in the function with specific values or expressions. This concept is key when evaluating a function at given parameter values to obtain the corresponding vector or point.

Transcript

00:01 Okay, we are going to evaluate, if possible, the vector valued functions shown.

00:08 So notice we have a vector valued function, and then we have four different parts where we're going to evaluate it at different places.

00:16 So r of zero means that we're placing in a zero for t.

00:21 Well, when you put a zero in cosine, you get a value of one, but when you put a zero in sine, you get a value of zero.

00:30 In the end, our answer is i.

00:33 You do not have to write the one, and you do not have to write any of your components that actually are zero.

00:40 So our r of pi over four would be placing a pi over four into our function.

00:47 So cosine of pi over four and sine a pi over four are square root of two over two.

00:53 The sign gets multiplied by a two, so you see just the square root of two.

00:57 Okay, for part c, we're going to be putting theta minus pi inside of cosine and sine as we write it.

01:07 So we have the cosine of theta minus pi in the i direction, plus that 2 sine of theta minus pi in the j direction.

01:16 So now, do you remember your sum and difference formulas? because this can be cleaned up dramatically if we remember those.

01:26 So remember, cosine is cosine, cosine, sine, sine, and the sign changes.

01:33 So we say cosine of theta times cosine of pi.

01:40 And what is the cosine of pi? it's just negative one.

01:42 So i'm going to throw a negative in there.

01:44 And then it would be minus the sign of both of them multiplied.

01:48 Remember, sine of pi is zero, so that piece can go away.

01:52 Now, when we do our, so that's our i component.

01:55 When we do our j component, it's similar.

01:58 For sine, sum, and difference, it goes sine, cosine, cosine, sine...

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