Problem 22 - Algebra and Trigonometry

Question

In Exercises 21–38, let. $$\mathbf{u}=2 \mathbf{i}-5 \mathbf{j}, \mathbf{v}=-3 \mathbf{i}+7 \mathbf{j}, \text { and } \mathbf{w}=-\mathbf{i}-6 \mathbf{j}$$ Find each specified vector or scalar.$$\mathbf{v}+\mathbf{w}$$

Step-by-Step Solution

Answer

The vector \(\mathbf{v} + \mathbf{w}\) is \(-4 \mathbf{i} + 1 \mathbf{j}\).

Step 1: Identify the components of the vectors

The first step is to identify the i (horizontal) and j (vertical) components of each vector. For vector \(\mathbf{v}\), the i-component is -3 and the j-component is +7. For vector \(\mathbf{w}\), the i-component is -1 and the j-component is -6.

Step 2: Add the i-components

The second step is to add the i-components of each vector, which corresponds to the horizontal direction. So, \(-3 \, (\text{i-component of}\, \mathbf{v}) - 1 \, (\text{i-component of} \, \mathbf{w}) = -4\).

Step 3: Add the j-components

The third step is to add the j-components of each vector, which corresponds to the vertical direction. So, \(7 \, (\text{j-component of} \, \mathbf{v}) - 6 \, (\text{j-component of} \, \mathbf{w}) = 1\).

Step 4: Write the resultant vector

The resultant vector, \(\mathbf{v} + \mathbf{w}\), is then the addition of the i and j components computed in steps 2 and 3, which gives us: \(-4 \mathbf{i} + 1 \mathbf{j}\).

Key Concepts

i and j components

Vectors are an essential component of mathematics, representing quantities that have both magnitude and direction. In two dimensions, vectors can be expressed in terms of their **i and j components**. The "i" component represents the horizontal part, while the "j" component denotes the vertical part of a vector.

For example, consider the vector \( \mathbf{v} = -3 \mathbf{i} + 7 \mathbf{j} \). Here, -3 is the coefficient of the **i-component**, indicating a movement in the horizontal direction to the left. Meanwhile, 7 is associated with the **j-component**, which moves upwards in the vertical direction. Understanding these components is crucial when performing operations like vector addition.

• **i-component**: Indicates movement along the horizontal axis (left or right).
• **j-component**: Indicates movement along the vertical axis (up or down).
• Together, they describe the vector's direction and magnitude in the 2D plane.

resultant vector

When combining two or more vectors, the outcome is known as a **resultant vector**. This vector represents the total effect of all individual vectors combined. Calculating the resultant vector involves adding the corresponding components of each vector.

For instance, when finding \( \mathbf{v} + \mathbf{w} \), you would add both the **i-components** and **j-components** separately:

• Add the i-components: \(-3 + (-1) = -4\).
• Add the j-components: \(7 + (-6) = 1\).

Therefore, the resultant vector for this particular problem is \(-4 \mathbf{i} + 1 \mathbf{j}\). This vector summarizes the combined effect of \( \mathbf{v} \) and \( \mathbf{w} \).

• Aggregates individual vector contributions into one.
• Useful in simplifying analysis by reducing multiple vectors into a single equivalent vector.
• Helps in visualizing the final direction and magnitude in scenarios like force or velocity addition.

horizontal and vertical components

Understanding the **horizontal and vertical components** is key to grasping vector behavior. Vectors often represent movements or forces, split into these two perpendicular directions.

**Horizontal components** parallel the x-axis, demonstrating left or right movement. They correspond to the i-component in two-dimensional vector notation. The influence of horizontal components can affect the range or coverage of a vector’s overall effect, like how far something might move horizontally.

**Vertical components** align with the y-axis, showing up or down movements corresponding to the j-component. These components can detail how high or low a vector-driven action might reach, such as elevating or dropping in space.

• **Horizontal (i) components** affect lateral movement.
• **Vertical (j) components** impact upward or downward movement.
• These components can be separately analyzed to deduce more about vector contributions in different directions.