Question

Are based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam. Find the unit vector in the same direction as \(\mathbf{v}=8 \mathbf{i}-15 \mathbf{j}\).

Short answer

\[ \mathbf{u} = \frac{8}{17}\mathbf{i} - \frac{15}{17}\mathbf{j} \]

Step-by-step solution

- Determine the magnitude of \(\mathbf{v}\)

The magnitude of a vector \(\mathbf{v} = a\mathbf{i} + b\mathbf{j}\) is given by the formula \(\|\mathbf{v}\| = \sqrt{a^2 + b^2}\). For \(\mathbf{v} = 8\mathbf{i} - 15\mathbf{j}\), we calculate: \[ \|\mathbf{v}\| = \sqrt{8^2 + (-15)^2} = \sqrt{64 + 225} = \sqrt{289} = 17 \].

- Divide each component of \(\mathbf{v}\) by its magnitude

To find the unit vector in the same direction as \(\mathbf{v}\), divide each component of \(\mathbf{v} = 8\mathbf{i} - 15\mathbf{j}\) by the magnitude \(17\). The unit vector \(\mathbf{u}\) is given by: \[ \mathbf{u} = \frac{1}{\|\mathbf{v}\|} \mathbf{v} = \frac{1}{17}(8\mathbf{i} - 15\mathbf{j}) = \frac{8}{17}\mathbf{i} - \frac{15}{17}\mathbf{j} \].

- Simplify the unit vector

Simplify the expression for the unit vector if necessary. In this case, the unit vector \(\mathbf{u}\) is: \[ \mathbf{u} = \frac{8}{17}\mathbf{i} - \frac{15}{17}\mathbf{j} \]. There is no further simplification required.

Key concepts

magnitude of a vector

To begin understanding vectors and their components, let's dive into the concept of the magnitude of a vector. The magnitude of a vector represents its length or size. Imagine a vector as an arrow pointing in space; the magnitude is the arrow's length.
To calculate the magnitude, use the formula: \ \ \ \ \ \[ \|\text{v}\| = \sqrt{a^2 + b^2} \] \ \ This formula works for any 2-dimensional vector \(\text{v} = a\text{i} + b\text{j}\). In our specific problem, we have the vector \(\text{v} = 8\text{i} - 15\text{j}\). Plugging in these values, we get: \ \[ \|\text{v}\| = \sqrt{8^2 + (-15)^2} = \sqrt{64 + 225} = \sqrt{289} = 17 \] \ Remember, the magnitude will always be a positive number, representing the vector's size without considering its direction.Mon tasks are standard and useful whenever we deal with vectors in different directions and magnitudes.

unit vector calculation

A unit vector is very helpful in physics and engineering because it simplifies calculations by standardizing the length. A unit vector always has a magnitude of 1, and it points in the same direction as the original vector. To find the unit vector for a given vector \(\text{v}\), you divide each component by the vector's magnitude.
Let's take our vector \(\text{v} = 8\text{i} - 15\text{j}\). We've already calculated its magnitude as 17.
Now, let's find the unit vector \(\text{u}\):
\[ \text{u} = \frac{1}{\|\text{v}\|} \text{v} = \frac{1}{17}(8\text{i} - 15\text{j}) = \frac{8}{17}\text{i} - \frac{15}{17}\text{j} \]
The unit vector is obtained by dividing the components of the initial vector (in this case, 8 and -15) by its magnitude (17). This process standardizes the vector's length to 1 while maintaining its direction. The result is: \( \text{u} = \frac{8}{17}\text{i} - \frac{15}{17}\text{j} \). This calculation helps to handle vectors efficiently in various applications.

vector components

Vectors can be broken down into their components, which are essentially the influences in the respective coordinate directions. For instance, the vector \( \text{v} = 8\text{i} - 15\text{j} \) can be understood as having two components: one in the x-direction (8) and one in the y-direction (-15). These components are important for understanding the vector's behavior in each direction.
This breaking down process is crucial when performing operations like addition, subtraction, and finding magnitudes. The i-component (\text{i}) represents the x-direction, and the j-component (\text{j}) represents the y-direction. For a quick recap:

• For any vector \( \text{v} = a\text{i} + b\text{j} \), 'a' is the i-component.
• 'b' is the j-component.

Knowing the components makes it easier to compute the vector's magnitude and find a unit vector in the same direction. Whenever you see a vector in the form \( \text{v} = a\text{i} + b\text{j} \), immediately identify 'a' and 'b' as the respective components. This foundational understanding will make all vector-related calculations much simpler.