Problem 17 - Algebra and Trigonometry

Question

15-18 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 result is -5.

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

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

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

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

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

Use the dot product formula \(\mathbf{a} \cdot \mathbf{b} = a_1b_1 + a_2b_2\), where \(\mathbf{a} = 3\mathbf{i} - 2\mathbf{j}\) and \(\mathbf{b} = \mathbf{i} + 4\mathbf{j}\):\[(3\mathbf{i} - 2\mathbf{j}) \cdot (\mathbf{i} + 4\mathbf{j}) = (3)(1) + (-2)(4) = 3 - 8 = -5.\]

Key Concepts

Understanding the Dot Product

The dot product is a way of multiplying two vectors that results in a scalar (a single number) rather than another vector. It's a fundamental operation in vector algebra. To compute the dot product of two vectors, you multiply their corresponding components and then sum the results.
For example, if we have two vectors \(\mathbf{a} = a_1\mathbf{i} + a_2\mathbf{j}\) and \(\mathbf{b} = b_1\mathbf{i} + b_2\mathbf{j}\), the dot product \(\mathbf{a} \cdot \mathbf{b}\) is calculated as:
\[ \mathbf{a} \cdot \mathbf{b} = a_1b_1 + a_2b_2 \]
This operation is very useful in determining angles between vectors and in finding the projection of one vector onto another.

• Yields a scalar value rather than a vector.
• Cannot find the direction but can find the magnitude relation between vectors.
• Used in physics to compute work done, where force and displacement are vector quantities.

The result of the dot product can tell you several things: if the result is positive, the vectors point roughly in the same direction. If it's negative, they point in opposite directions. If zero, the vectors are perpendicular.

Explaining Vector Addition

Vector addition combines two or more vectors to result in another vector. This is done by adding the corresponding components of each vector. It's quite a straightforward operation and visually, you can think of it as connecting two vectors tip to tail and drawing a vector from the start of the first to the end of the second.
Mathematically, if you have \(\mathbf{u} = u_1\mathbf{i} + u_2\mathbf{j}\) and \(\mathbf{v} = v_1\mathbf{i} + v_2\mathbf{j}\), their addition is:
\[ \mathbf{u} + \mathbf{v} = (u_1 + v_1)\mathbf{i} + (u_2 + v_2)\mathbf{j} \]
Vector addition follows the commutative property, meaning \(\mathbf{u} + \mathbf{v} = \mathbf{v} + \mathbf{u}\), and the associative property, meaning \((\mathbf{u} + \mathbf{v}) + \mathbf{w} = \mathbf{u} + (\mathbf{v} + \mathbf{w})\).

• Resulting vector is the diagonal of the parallelogram formed.
• Used in physics to sum forces, velocities, and other vector quantities.
• Helps visualize combined effects of vectors.

Vector addition is not just limited to physical quantities. It can also apply to any form of directional data.

The Concept of Vector Subtraction

Vector subtraction is similar to vector addition, but instead of adding, you subtract the components of the vectors. If you have two vectors, \(\mathbf{u} = u_1\mathbf{i} + u_2\mathbf{j}\) and \(\mathbf{v} = v_1\mathbf{i} + v_2\mathbf{j}\), then the subtraction \(\mathbf{u} - \mathbf{v}\) is calculated as:
\[ \mathbf{u} - \mathbf{v} = (u_1 - v_1)\mathbf{i} + (u_2 - v_2)\mathbf{j} \]
Subtracting a vector is akin to reversing its direction and then performing vector addition.

• Used to determine the relative position change of one vector with respect to another.
• Can help in understanding displacement or changes in velocity.
• Visually represented by creating a vector from the tip of the vector being subtracted to the tip of the vector doing the subtraction.

This operation is essential in many topics, such as physics, to find relative velocities or to compute how one force negates another.