Question

If \(\mathbf{v}=a_{1} \mathbf{i}+b_{1} \mathbf{j}\) and \(\mathbf{w}=a_{2} \mathbf{i}+b_{2} \mathbf{j}\) are two vectors, then the _______ ______is defined as \(\mathbf{v} \cdot \mathbf{w}=a_{1} a_{2}+b_{1} b_{2}\).

Short answer

'dot product'.

Step-by-step solution

Rewrite the Given Vectors

The problem provides two vectors \(\mathbf{v} = a_1\mathbf{i} + b_1\mathbf{j}\) and \(\mathbf{w} = a_2\mathbf{i} + b_2\mathbf{j}\). These vectors are in component form.

Understand the Dot Product Definition

The dot product of two vectors \(\mathbf{v}\) and \(\mathbf{w}\) is a scalar obtained by multiplying corresponding components and adding them together. This operation is typically represented as \(\mathbf{v} \cdot \mathbf{w}\).

Apply the Dot Product Formula

For the vectors \(\mathbf{v} = a_1\mathbf{i} + b_1\mathbf{j}\) and \(\mathbf{w} = a_2\mathbf{i} + b_2\mathbf{j}\), the dot product is calculated as \(\mathbf{v} \cdot \mathbf{w} = a_1 a_2 + b_1 b_2\).

Define the Term

The term used to describe the operation \(\mathbf{v} \cdot \mathbf{w} = a_1 a_2 + b_1 b_2\) is called the 'dot product.'

Key concepts

headline of the respective core concept

Vectors are fundamental entities in mathematics and physics. A vector is a quantity that has both magnitude and direction. We often represent vectors in a coordinate system using the notation \( \mathbf{i} \) and \( \mathbf{j} \) as unit vectors along the x and y axes, respectively.
An example of a vector is \( \mathbf{v} = a_1\mathbf{i} + b_1\mathbf{j} \), where \( a_1 \) and \( b_1 \) are the components of the vector along the x and y axes. These components describe how far and in which direction the vector extends from the origin in each dimension.
Understanding vectors is crucial because they help describe physical quantities like force, velocity, and displacement.

dot product definition

The dot product is a specific way to multiply two vectors. It results in a scalar quantity rather than a vector, which means the result has magnitude but no direction.
For two vectors \( \mathbf{v} = a_1\mathbf{i} + b_1\mathbf{j} \) and \( \mathbf{w} = a_2\mathbf{i} + b_2\mathbf{j} \), the dot product is defined as follows:
\[ \mathbf{v} \cdot \mathbf{w} = a_1 a_2 + b_1 b_2 \]
Here, \( \mathbf{v} \cdot \mathbf{w} \) (read as 'v dot w') means you multiply the corresponding components of the vectors and then sum these products. The resultant scalar can tell us about the relationship between the two vectors — for example, whether they are perpendicular or if one vector projects onto another.

scalar multiplication

Scalar multiplication involves multiplying a vector by a single number (called a 'scalar'), which scales the vector's magnitude without changing its direction.
Given a vector \( \mathbf{v} = a_1\mathbf{i} + b_1\mathbf{j} \) and a scalar \( k \), the result of the scalar multiplication is:
\[ k \mathbf{v} = k(a_1\mathbf{i} + b_1\mathbf{j}) = (k a_1) \mathbf{i} + (k b_1) \mathbf{j} \]
For instance, if \( k = 2 \) and \( \mathbf{v} = 3\mathbf{i} + 4\mathbf{j} \), then \( 2 \mathbf{v} = 6\mathbf{i} + 8\mathbf{j} \). The vector effectively grows in magnitude by a factor of 2.

vector operations

Working with vectors involves various operations such as addition, subtraction, and the dot product.

• **Addition**: Adding two vectors \( \mathbf{v} = a_1 \mathbf{i} + b_1 \mathbf{j} \) and \( \mathbf{w} = a_2 \mathbf{i} + b_2 \mathbf{j} \) results in a new vector: \[ \mathbf{v} + \mathbf{w} = (a_1 + a_2) \mathbf{i} + (b_1 + b_2) \mathbf{j} \]
• **Subtraction**: Subtracting two vectors \( \mathbf{v} \) and \( \mathbf{w} \) follows similarly, resulting in: \[ \mathbf{v} - \mathbf{w} = (a_1 - a_2) \mathbf{i} + (b_1 - b_2) \mathbf{j} \]
• **Dot Product**: As detailed in the previous section, the dot product is a form of multiplication that results in a scalar: \[ \mathbf{v} \cdot \mathbf{w} = a_1 a_2 + b_1 b_2 \]

Understanding these operations is fundamental to solving more complex problems involving vectors in physics and engineering.