91Ó°ÊÓ

The position vector is a way to represent a point's location in a given space, like a snapshot of where you are at any moment. In our exercise, the position vector \( \mathbf{r}(t) \) is written as \( \langle 30 \cos(2t), 40 \sin(2t), 20t + 3t^2 \rangle \). This indicates that:

• \( x(t) = 30 \cos(2t) \) - the x-component depends on the cosine function and changes as \( 2t \) varies.
• \( y(t) = 40 \sin(2t) \) - the y-component relies on the sine function, also changing with \( 2t \).
• \( z(t) = 20t + 3t^2 \) - the z-component is a polynomial, showing how the position changes with time.

Evaluating these at \( t=2 \), we get \( \mathbf{r}(2) = \langle 30 \cos(4), 40 \sin(4), 46 \rangle \), which gives a specific point in space at that moment.

Think of the velocity vector as the position vector's rate of change, like how fast and in which direction you're moving at any given time. By differentiating \( \mathbf{r}(t) \) with respect to \( t \), we obtain the velocity vector \( \mathbf{v}(t) = \langle -60 \sin(2t), 80 \cos(2t), 20 + 6t \rangle \). Each component shows:

• Rate of change of x: \( -60 \sin(2t) \) - how fast the x-position is changing due to the sin function.
• Rate of change of y: \( 80 \cos(2t) \) - reflecting the y-component change due to the cos function.
• Rate of change of z: \( 20 + 6t \) - this direct arithmetic expression shows constant and linearly increasing speed in the z-direction.

For \( t = 2 \), \( \mathbf{v}(2) = \langle -60 \sin(4), 80 \cos(4), 32 \rangle \), representing the velocity vector at that specific time.

The acceleration vector describes how the velocity vector changes with time, either speeding up or slowing down. By taking the derivative of \( \mathbf{v}(t) \), we find \( \mathbf{a}(t) = \langle -120 \cos(2t), -160 \sin(2t), 6 \rangle \). These components indicate:

• The x-component \( -120 \cos(2t) \) tells us how fast the x-related velocity increases or decreases.
• The y-component \( -160 \sin(2t) \) indicates similar for the y-related velocity.
• The constant z-component \( 6 \) represents constant acceleration in the z-direction.

At \( t = 2 \), the acceleration vector is \( \mathbf{a}(2) = \langle -120 \cos(4), -160 \sin(4), 6 \rangle \), detailing how the velocity changes at this precise time.

The dot product is a mathematical operation that combines two vectors and provides a scalar, often used to find angles between vectors. Calculating the dot product of two vectors, such as \( \mathbf{r}(2) \) and \( \mathbf{v}(2) \), involves multiplying corresponding components and then summing them up. In this context, it's represented as:\[\mathbf{r}(2) \cdot \mathbf{v}(2) = 30 \cos(4)(-60 \sin(4)) + 40 \sin(4)(80 \cos(4)) + 46(32)\]The result helps determine the angle between the position and velocity vectors using the formula:\[\cos(\theta_1) = \frac{\mathbf{r}(2) \cdot \mathbf{v}(2)}{\|\mathbf{r}(2)\| \|\mathbf{v}(2)\|}\]This formula finds \( \theta_1 \), the angle between these vectors at \( t = 2 \mathrm{s} \). The same process applies to find \( \theta_2 \) for \( \mathbf{r}(2) \) and \( \mathbf{a}(2) \).

Trigonometric functions like sine and cosine are fundamental in describing oscillatory motion, which can be seen in this exercise. They help model periodic phenomena such as the alternating components of the position and velocity vectors:

• Cosine function \( \cos(2t) \) emphasizes how the position or velocity cycles with a specific frequency determined by the coefficient 2.
• Sine function \( \sin(2t) \) similarly cycles, but with a phase shift indicated by its periodic nature.

These functions are practical for breaking down complex movements into simple harmonic parts and are essential for calculating exact values, like knowing the exact position or velocity at a given time \( t \). By understanding these functions, the behavior of vectors in multiple dimensions becomes clear, aiding in the problem's solution.