Problem 48 - Calculus Early Transcendentals: Pearson New International Edition

Question

In Problems 45-50, give a proof of the indicated property for two-dimensional vectors. Use \(\mathbf{u}=\left\langle u_{1}, u_{2}\right\rangle, \mathbf{v}=\left\langle v_{1}, v_{2}\right\rangle\), and \(\mathbf{w}=\left\langle w_{1}, w_{2}\right\rangle\). $$ \mathbf{u} \cdot(\mathbf{v}+\mathbf{w})=\mathbf{u} \cdot \mathbf{v}+\mathbf{u} \cdot \mathbf{w} $$

Step-by-Step Solution

Answer

The distributive property for dot products is proved by expanding both sides to \(u_1v_1 + u_2v_2 + u_1w_1 + u_2w_2\).

Step 1: Identify the Property

The property to be proved is the distributive property of the dot product for two-dimensional vectors. This property states that the dot product of one vector with the sum of two others is equal to the dot product of the first vector with each of the other vectors, summed together.

Step 2: Express Vectors in Component Form

Write down the vectors in component form: \(\mathbf{u}=\langle u_1, u_2\rangle\), \(\mathbf{v}=\langle v_1, v_2\rangle\), and \(\mathbf{w}=\langle w_1, w_2\rangle\).

Step 3: Compute the Left Side

Calculate the left side of the equation \(\mathbf{u} \cdot (\mathbf{v}+\mathbf{w})\). First, find \(\mathbf{v} + \mathbf{w} = \langle v_1 + w_1, v_2 + w_2 \rangle\). Then compute the dot product: \(\mathbf{u} \cdot (\mathbf{v} + \mathbf{w}) = u_1(v_1 + w_1) + u_2(v_2 + w_2)\).

Step 4: Simplify the Left Side

Distribute \(u_1\) and \(u_2\) into the expression: \(u_1v_1 + u_1w_1 + u_2v_2 + u_2w_2\).

Step 5: Compute the Right Side

Calculate each part of the right side separately: \(\mathbf{u} \cdot \mathbf{v} = u_1v_1 + u_2v_2\), and \(\mathbf{u} \cdot \mathbf{w} = u_1w_1 + u_2w_2\). Add them together to get \(u_1v_1 + u_2v_2 + u_1w_1 + u_2w_2\).

Step 6: Verify Equality

Compare the expressions obtained in Steps 4 and 5. Both are \(u_1v_1 + u_2v_2 + u_1w_1 + u_2w_2\), thus proving the distributive property.

Key Concepts

Dot Product

The dot product, also known as the scalar product, is a fundamental operation in vector calculus. It involves two vectors and yields a single number, or scalar.
If you have two two-dimensional vectors such as \( \mathbf{u} = \langle u_1, u_2 \rangle \) and \( \mathbf{v} = \langle v_1, v_2 \rangle \), their dot product is calculated as follows:

• Multiply the corresponding components of the vectors together. This means you multiply \( u_1 \) with \( v_1 \) and \( u_2 \) with \( v_2 \).
• Then, sum these products to obtain the dot product: \( \mathbf{u} \cdot \mathbf{v} = u_1v_1 + u_2v_2 \).

The result of the dot product is a scalar, not a vector. This characteristic makes it different from other vector operations. Understanding the dot product is crucial as it is used in various applications including equations of lines, planes, and determining angles between vectors.

Distributive Property

The distributive property is an important concept in mathematics that also applies to the dot product of vectors.
For vectors, the distributive property states that if you have a vector \( \mathbf{u} \) and two other vectors \( \mathbf{v} \) and \( \mathbf{w} \), the expression \( \mathbf{u} \cdot (\mathbf{v} + \mathbf{w}) \) can be expanded using the distributive property:

• First, compute the vector sum \( \mathbf{v} + \mathbf{w} \).
• Then apply the dot product: \( \mathbf{u} \cdot (\mathbf{v} + \mathbf{w}) = \mathbf{u} \cdot \mathbf{v} + \mathbf{u} \cdot \mathbf{w} \).

This means you're allowed to perform the dot product separately on each vector, then add those results. This property helps simplify calculations and is used to prove various vector identities.

Two-Dimensional Vectors

Vectors are fundamental in representing quantities that have both a magnitude and a direction. In the context of two-dimensional vectors, each vector can be described using two components or coordinates.
A vector \( \mathbf{u} \) in two-dimensional space is typically written as \( \langle u_1, u_2 \rangle \).

• The first component, \( u_1 \), represents the vector's movement or influence along the horizontal axis, also known as the x-axis.
• The second component, \( u_2 \), represents the vector's movement along the vertical axis, or y-axis.

These vectors can be visualized as arrows in the plane pointing from the origin to the point \( (u_1, u_2) \). Two-dimensional vectors are used extensively in graphics, physics, engineering, and any field involving planar motion. Understanding their properties, including operations like addition, subtraction, and the dot product, is crucial to mastering the subject of vector calculus.