91Ó°ÊÓ

In vector calculus, the position vector is a fundamental concept that allows us to describe the position of a point in space with respect to a fixed origin. Think of it as a directed line segment from the origin of your coordinate system to the point in question. This is mathematically expressed as: \[ \mathbf{r}(t) = x(t) \mathbf{i} + y(t) \mathbf{j} + z(t) \mathbf{k} \]where \(x(t)\), \(y(t)\), and \(z(t)\) indicate the coordinates of the point in space as functions of time. In the original exercise, the position vector is given as \(\mathbf{r}(t) = t \mathbf{i} + t^2 \mathbf{j} + t^3 \mathbf{k}\). This shows that as time progresses, the point's location changes in three-dimensional space, following a certain path. The position vector helps us track how this point moves over time, allowing for calculations regarding its speed and direction.

The velocity vector is derived by differentiating the position vector with respect to time. It provides vital information about the rate of change of the position vector. Essentially, the velocity vector describes both how fast and in which direction the particle is moving. Mathematically, it is represented as:\[ \mathbf{v}(t) = \frac{d\mathbf{r}(t)}{dt} = \frac{d}{dt}(x(t) \mathbf{i} + y(t) \mathbf{j} + z(t) \mathbf{k}) \]In our example, by differentiating \(\mathbf{r}(t) = t \mathbf{i} + t^2 \mathbf{j} + t^3 \mathbf{k}\), we get the velocity vector:\[ \mathbf{v}(t) = \mathbf{i} + 2t \mathbf{j} + 3t^2 \mathbf{k} \]At \(t = 1\), the velocity vector is \(\mathbf{v}(1) = \mathbf{i} + 2 \mathbf{j} + 3 \mathbf{k}\), describing the velocity of the particle at this specific moment. This vector is crucial for understanding how the particle moves along its trajectory.

The acceleration vector is a second derivative level calculation following the velocity vector. It indicates how the velocity of the object changes over time. This not only involves changes in the speed but also how the direction of the velocity changes. Formally, the acceleration vector is represented by:\[ \mathbf{a}(t) = \frac{d\mathbf{v}(t)}{dt} \]Applying this to our example, the velocity vector \(\mathbf{v}(t) = \mathbf{i} + 2t \mathbf{j} + 3t^2 \mathbf{k}\) gives an acceleration vector of:\[ \mathbf{a}(t) = 2 \mathbf{j} + 6t \mathbf{k} \]When \(t = 1\), the acceleration vector becomes \(\mathbf{a}(1) = 2 \mathbf{j} + 6 \mathbf{k}\). This reveals the vector's magnitude and direction of acceleration at that instant—a crucial step in finding both tangential and normal components of the acceleration.

The magnitude of a vector represents its length or size, regardless of its direction. It is fundamentally important when studying vectors to understand how strong or significant their impact is. The magnitude for a vector \( \mathbf{v} = a \mathbf{i} + b \mathbf{j} + c \mathbf{k} \) is found using:\[ \|\mathbf{v}\| = \sqrt{a^2 + b^2 + c^2} \]In the exercise, calculating the magnitude of the velocity vector \(\mathbf{v}(1) = \mathbf{i} + 2 \mathbf{j} + 3 \mathbf{k}\) gives:\[ \|\mathbf{v}(1)\| = \sqrt{1^2 + 2^2 + 3^2} = \sqrt{14} \]This numerical value is essential for further calculations, such as determining the tangential and normal components of the acceleration. It helps in understanding how pronounced the velocity of the particle is at the given instance.

The dot product is an operation that takes two vectors and returns a scalar quantity. This mathematical operation is essential when projecting one vector onto another or when finding angles between vectors. Mathematically, it is expressed as:\[ \mathbf{a} \cdot \mathbf{b} = a_1b_1 + a_2b_2 + a_3b_3 \]In our task, the dot product is used to find the tangential component of acceleration. Following the example, for vectors \(\mathbf{a}(1) = 2 \mathbf{j} + 6 \mathbf{k}\) and \(\mathbf{v}(1) = \mathbf{i} + 2 \mathbf{j} + 3 \mathbf{k}\), the dot product is:\[ \mathbf{a}(1) \cdot \mathbf{v}(1) = 0 \times 1 + 2 \times 2 + 6 \times 3 = 22 \]This value contributes to finding the tangential component of the acceleration by measuring how much of the acceleration vector aligns with the velocity vector.