Problem 22 - Algebra and Trigonometry

Question

Find the indicated quantity, assuming \(\mathbf{u}=2 \mathbf{i}+\mathbf{j}, \mathbf{v}=\mathbf{i}-3 \mathbf{j},\) and \(\mathbf{w}=3 \mathbf{i}+4 \mathbf{j}\) $$ \mathbf{u} \cdot(\mathbf{v}+\mathbf{w}) $$

Step-by-Step Solution

Answer

The result of \( \mathbf{u} \cdot (\mathbf{v} + \mathbf{w}) \) is 9.

Step 1: Find \( \mathbf{v} + \mathbf{w} \)

To find \( \mathbf{v} + \mathbf{w} \), add the corresponding components of the vectors \( \mathbf{v} \) and \( \mathbf{w} \).\[ \mathbf{v} + \mathbf{w} = (\mathbf{i} - 3\mathbf{j}) + (3\mathbf{i} + 4\mathbf{j}) = (1 + 3)\mathbf{i} + (-3 + 4)\mathbf{j} \] \[ \mathbf{v} + \mathbf{w} = 4\mathbf{i} + \mathbf{j} \]

Step 2: Calculate \( \mathbf{u} \cdot (\mathbf{v} + \mathbf{w}) \)

Use the dot product formula \( \mathbf{a} \cdot \mathbf{b} = a_1b_1 + a_2b_2 \). Substitute \( \mathbf{u} = 2\mathbf{i} + \mathbf{j} \) and \( \mathbf{v} + \mathbf{w} = 4\mathbf{i} + \mathbf{j} \).\[ \mathbf{u} \cdot (\mathbf{v} + \mathbf{w}) = (2\mathbf{i} + \mathbf{j}) \cdot (4\mathbf{i} + \mathbf{j}) \] \[ = 2 \cdot 4 + 1 \cdot 1 \] \[ = 8 + 1 \] \[ = 9 \]

Key Concepts

Vector Addition

Vector addition is a fundamental operation used to combine two or more vectors. When we add vectors, we perform the operation by adding their corresponding components. For any two vectors, \( \mathbf{a} = a_1 \mathbf{i} + a_2 \mathbf{j} \) and \( \mathbf{b} = b_1 \mathbf{i} + b_2 \mathbf{j} \), their sum \( \mathbf{a} + \mathbf{b} \) can be found by:

• Adding the \( \mathbf{i} \) components: \( a_1 + b_1 \)
• Adding the \( \mathbf{j} \) components: \( a_2 + b_2 \)

This gives the resultant vector: \( \mathbf{a} + \mathbf{b} = (a_1 + b_1) \mathbf{i} + (a_2 + b_2) \mathbf{j} \).
In the provided exercise, you added vector \( \mathbf{v} = \mathbf{i} - 3 \mathbf{j} \) and vector \( \mathbf{w} = 3 \mathbf{i} + 4 \mathbf{j} \). The operation involves adding their respective components:

• The \( \mathbf{i} \) components: \( 1 + 3 = 4 \)
• The \( \mathbf{j} \) components: \( -3 + 4 = 1 \)

This results in the vector \( 4 \mathbf{i} + \mathbf{j} \), which is the sum of \( \mathbf{v} \) and \( \mathbf{w} \).
Thus, vector addition helps in combining different vectors to form a resultant vector.

Vector Components

Vectors are quantities that have both magnitude and direction, and they can be expressed in terms of their components. The components of a vector are the projections of the vector along the coordinate axes. In two-dimensional space, any vector \( \mathbf{a} \) can be represented as \( a_1 \mathbf{i} + a_2 \mathbf{j} \), where:

• \( a_1 \) is the component along the \( \mathbf{i} \) (horizontal) axis.
• \( a_2 \) is the component along the \( \mathbf{j} \) (vertical) axis.

The vector components allow us to break down complex vector operations into more manageable calculations.
In the example given, you work with the vectors \( \mathbf{u}, \mathbf{v}, \) and \( \mathbf{w} \), each expressed in terms of \( \mathbf{i} \) and \( \mathbf{j} \) components:

• \( \mathbf{u} = 2 \mathbf{i} + \mathbf{j} \) with components \( (2, 1) \).
• \( \mathbf{v} = \mathbf{i} - 3 \mathbf{j} \) with components \( (1, -3) \).
• \( \mathbf{w} = 3 \mathbf{i} + 4 \mathbf{j} \) with components \( (3, 4) \).

Understanding the components of a vector is crucial for performing operations like addition, subtraction, and in this case the dot product.

Vector Operations

Vector operations allow us to manipulate vectors mathematically. One of the fundamental operations is the dot product, which measures the extent to which two vectors are parallel. The dot product of two vectors \( \mathbf{a} = a_1 \mathbf{i} + a_2 \mathbf{j} \) and \( \mathbf{b} = b_1 \mathbf{i} + b_2 \mathbf{j} \) is calculated using the formula:

• \( \mathbf{a} \cdot \mathbf{b} = a_1b_1 + a_2b_2 \)

This operation yields a scalar, not a vector.
In the exercise, you find the dot product of \( \mathbf{u} \) and \( \mathbf{v} + \mathbf{w} \), where \( \mathbf{u} = 2 \mathbf{i} + \mathbf{j} \) and \( \mathbf{v} + \mathbf{w} = 4 \mathbf{i} + \mathbf{j} \). You calculated:

• \( 2 \cdot 4 + 1 \cdot 1 = 8 + 1 = 9 \)

The result, 9, indicates the level of alignment between vector \( \mathbf{u} \) and the sum of \( \mathbf{v} \) and \( \mathbf{w} \). The dot product is useful in various applications, such as finding angles between vectors and projecting one vector onto another.