Problem 23 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

The value is \(-5\).

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

To find \(\mathbf{u}+\mathbf{v}\), add the corresponding components of \(\mathbf{u}\) and \(\mathbf{v}\). For the \(\mathbf{i}\) components: \(2 + 1 = 3\). For the \(\mathbf{j}\) components: \(1 - 3 = -2\). Thus, \(\mathbf{u}+\mathbf{v} = 3\mathbf{i} - 2\mathbf{j}\).

Step 2: Find \(\mathbf{u}-\mathbf{v}\)

To find \(\mathbf{u}-\mathbf{v}\), subtract the components of \(\mathbf{v}\) from \(\mathbf{u}\). For the \(\mathbf{i}\) components: \(2 - 1 = 1\). For the \(\mathbf{j}\) components: \(1 - (-3) = 4\). Therefore, \(\mathbf{u}-\mathbf{v} = \mathbf{i} + 4\mathbf{j}\).

Step 3: Compute the Dot Product \((\mathbf{u}+\mathbf{v}) \cdot (\mathbf{u}-\mathbf{v})\)

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 as \(a_1b_1 + a_2b_2\). Here, \(\mathbf{a} = 3\mathbf{i} - 2\mathbf{j}\) and \(\mathbf{b} = \mathbf{i} + 4\mathbf{j}\). Thus, the dot product is \[3 \times 1 + (-2) \times 4 = 3 - 8 = -5\].

Step 4: Conclusion: Final Result

The value of \((\mathbf{u}+\mathbf{v}) \cdot (\mathbf{u}-\mathbf{v})\) is \(-5\).

Key Concepts

Vector Addition

Vector addition is a fundamental operation where we combine two or more vectors to form a single resultant vector. Each vector has components that align along coordinate axes, like the i (horizontal) and j (vertical) axes in two dimensions. To add vectors, you simply add their corresponding components.
For example, if you have vectors \( \mathbf{u} = 2 \mathbf{i} + \mathbf{j} \) and \( \mathbf{v} = \mathbf{i} - 3 \mathbf{j} \), the vector addition \( \mathbf{u} + \mathbf{v} \) is done as follows:

• Add the i-components: \(2 + 1 = 3 \)
• Add the j-components: \(1 + (-3) = -2 \)

Thus, the resultant vector becomes \( 3\mathbf{i} - 2\mathbf{j} \). This resultant vector shows the combined effect of both vectors in relation to direction and magnitude.

Vector Subtraction

Vector subtraction involves finding the difference between two vectors. Much like addition, this process also involves the subtraction of corresponding components. If you consider vectors \( \mathbf{u} = 2 \mathbf{i} + \mathbf{j} \) and \( \mathbf{v} = \mathbf{i} - 3 \mathbf{j} \), vector subtraction is executed as follows:

• Subtract the i-components: \(2 - 1 = 1 \)
• Subtract the j-components: \(1 - (-3) = 4 \)

The result, \( \mathbf{i} + 4\mathbf{j} \), signifies the difference between these two vectors. This vector shows how much \( \mathbf{u} \) differs from \( \mathbf{v} \) in both direction and magnitude.

Algebraic Vectors

Algebraic vectors are vectors expressed in terms of their components along the coordinate axes, typically using unit vectors like \( \mathbf{i} \) and \( \mathbf{j} \) in two-dimensional space. These components represent the vector’s influence along the horizontal and vertical directions, respectively.
Understanding algebraic vectors facilitates operations like addition, subtraction, and the dot product, by focusing on numeric manipulation of these components. The algebraic approach makes visualization more precise by breaking down vectors into identifiable, numerical terms.
For instance, to calculate the dot product of two algebraic vectors \( \mathbf{a} = 3 \mathbf{i} - 2 \mathbf{j} \) and \( \mathbf{b} = \mathbf{i} + 4 \mathbf{j} \), we perform:

• Multiply respective i-components: \(3 \times 1 = 3 \)
• Multiply respective j-components: \((-2) \times 4 = -8 \)

Adding these products gives the total sum for the dot product: \(3 - 8 = -5\). This result reflects the interaction of two vectors' magnitudes and directions in numerical form.