Problem 107 - Algebra and Trigonometry

Question

For \(\mathbf{v}=2 \mathbf{i}-\mathbf{j}\) and \(\mathbf{w}=\mathbf{i}+2 \mathbf{j},\) find the dot product \(\mathbf{v} \cdot \mathbf{w}\).

Step-by-Step Solution

Answer

The dot product is 0.

Step 1 - Write down the given vectors

The given vectors are: \( \mathbf{v} = 2 \mathbf{i} - \mathbf{j} \) \( \mathbf{w} = \mathbf{i} + 2 \mathbf{j} \)

Step 2 - Use the dot product formula

The dot product formula for vectors \( \mathbf{v} = a_1 \mathbf{i} + b_1 \mathbf{j} \) and \( \mathbf{w} = a_2 \mathbf{i} + b_2 \mathbf{j} \) is: \( \mathbf{v} \cdot \mathbf{w} = a_1a_2 + b_1b_2 \)

Step 3 - Substitute the components into the formula

Identify the components from the given vectors: For \( \mathbf{v} \): \( a_1 = 2 \), \( b_1 = -1 \) For \( \mathbf{w} \): \( a_2 = 1 \), \( b_2 = 2 \) Substitute these values into the dot product formula: \( \mathbf{v} \cdot \mathbf{w} = (2)(1) + (-1)(2) \)

Step 4 - Perform the arithmetic operations

Calculate the result: \( \mathbf{v} \cdot \mathbf{w} = 2 \times 1 + (-1) \times 2 = 2 - 2 = 0 \)

Key Concepts

vectors

In mathematics and physics, a vector is a quantity that has both magnitude and direction. Vectors are usually represented in a coordinate system with two or more components.
In two-dimensional space, vectors are often written in the form \(\text{{v}} = a \text{{i}} + b \text{{j}}\), where \(\text{{i}}\) and \(\text{{j}}\) represent unit vectors along the x- and y-axes, respectively.
A vector can describe many physical quantities, such as position, velocity, force, and more.
One of the key operations involving vectors is the dot product, which measures how much one vector extends in the direction of another.

vector components

To better understand vectors, it's essential to break them down into their components. The components of a vector are its projections along the coordinate axes.
For example, given a vector \(\text{{v}} = 2 \text{{i}} - \text{{j}}\), the components are:

• \(2 \text{{i}}\) - This is the x-component of the vector.
• \(- \text{{j}}\) - This is the y-component of the vector.

Understanding the components of a vector helps in performing vector operations like addition, subtraction, and the dot product.
In our exercise, we identified the components of the vectors \(\text{{v}}\) and \(\text{{w}}\):
\(\text{{v}} = 2 \text{{i}} - \text{{j}}\) has components \((2, -1)\)
\(\text{{w}} = \text{{i}} + 2 \text{{j}}\) has components \((1, 2)\).

dot product formula

The dot product (also known as the scalar product) is a way to multiply two vectors to obtain a scalar quantity.
The formula for the dot product of two vectors \(\text{{v}} = a_1 \text{{i}} + b_1 \text{{j}}\) and \(\text{{w}} = a_2 \text{{i}} + b_2 \text{{j}}\) is:
\[ \text{{v}} \cdot \text{{w}} = a_1a_2 + b_1b_2 \]
This formula essentially multiplies the respective components of the two vectors and sums the results.
Using this formula helps to find out how much one vector is in the direction of another and it's useful in many applications, like physics and engineering.
In our exercise, we used the dot product formula to find \(\text{{v}} \cdot \text{{w}} = 2 \cdot 1 + (-1) \cdot 2 = 2 - 2 = 0\). This indicates that the vectors \(\text{{v}}\) and \(\text{{w}}\) are orthogonal (perpendicular) to each other.

vector arithmetic

Vector arithmetic includes a variety of operations that can be performed on vectors. These operations include addition, subtraction, and multiplication.

• Vector addition: This involves adding the corresponding components of two vectors \(\text{{v}}\) and \(\text{{w}}\). For example, \(\text{{v}} = 2 \text{{i}} - \text{{j}}\) and \(\text{{w}} = \text{{i}} + 2 \text{{j}}\) can be added to get \(\text{{v + w}} = (2 + 1) \text{{i}} + (-1 + 2) \text{{j}} = 3 \text{{i}} + \text{{j}}\).
• Vector subtraction: This involves subtracting the corresponding components of two vectors. For example, \(\text{{v - w}} = (2 - 1) \text{{i}} + (-1 - 2) \text{{j}} = \text{{i}} - 3 \text{{j}}\).
• Scalar multiplication: This involves multiplying each component of a vector by a scalar. For instance, multiplying \(\text{{v}} = 2 \text{{i}} - \text{{j}}\) by 3 gives \((3 \times 2) \text{{i}} + (3 \times -1) \text{{j}} = 6 \text{{i}} - 3 \text{{j}}\).

Understanding these operations is crucial for more advanced studies and applications involving vectors.