Problem 47 - 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 \(\mathbf{w}=\left\langle w_{1}, w_{2}\right\rangle\). $$ c(\mathbf{u} \cdot \mathbf{v})=(c \mathbf{u}) \cdot \mathbf{v} $$

Step-by-Step Solution

Answer

Both sides simplify to \(cu_1 \cdot v_1 + cu_2 \cdot v_2\), proving the property.

Step 1: Understand the Dot Product

The dot product for two-dimensional vectors \(\mathbf{u} = \langle u_1, u_2 \rangle\) and \(\mathbf{v} = \langle v_1, v_2 \rangle\) is calculated as \(\mathbf{u} \cdot \mathbf{v} = u_1 \cdot v_1 + u_2 \cdot v_2\). We need to show that multiplying the dot product by a scalar \(c\) is equivalent to multiplying \(\mathbf{u}\) by \(c\) first and then taking the dot product with \(\mathbf{v}\).

Step 2: Simplify the Left Side

The left side of the equation is \(c(\mathbf{u} \cdot \mathbf{v})\). Using the definition of the dot product, this becomes \(c(u_1 \cdot v_1 + u_2 \cdot v_2)\), which can be distributed to get \(cu_1 \cdot v_1 + cu_2 \cdot v_2\).

Step 3: Simplify the Right Side

The right side of the equation is \((c \mathbf{u}) \cdot \mathbf{v}\). First, multiply \(\mathbf{u}\) by the scalar \(c\) to get \(c \mathbf{u} = \langle cu_1, cu_2 \rangle\). Then, take the dot product with \(\mathbf{v}\): \(\langle cu_1, cu_2 \rangle \cdot \langle v_1, v_2 \rangle = cu_1 \cdot v_1 + cu_2 \cdot v_2\).

Step 4: Compare Both Sides

From Steps 2 and 3, we saw that the left side simplifies to \(cu_1 \cdot v_1 + cu_2 \cdot v_2\) and the right side also simplifies to \(cu_1 \cdot v_1 + cu_2 \cdot v_2\). Since both sides are equal, the property \(c(\mathbf{u} \cdot \mathbf{v}) = (c \mathbf{u}) \cdot \mathbf{v}\) is proven.

Key Concepts

Dot Product

The dot product is a key concept in vector calculus. It takes two vectors and results in a scalar, essentially a single number. For two-dimensional vectors like \(\mathbf{u} = \langle u_1, u_2 \rangle\) and \(\mathbf{v} = \langle v_1, v_2 \rangle\), the dot product is calculated as:

• \(\mathbf{u} \cdot \mathbf{v} = u_1 \cdot v_1 + u_2 \cdot v_2\)

This operation combines corresponding components of the vectors, multiplying them together and then adding the results. The dot product is important because it can tell you about the angle between two vectors, along with other vector alignment properties. If the dot product is positive, the vectors point in broadly the same direction. If it's negative, they are pointing in opposite directions. When it's zero, the vectors are perpendicular to each other, meaning they form a right angle.

Scalar Multiplication

Scalar multiplication is a simple yet crucial operation in vector mathematics. It involves multiplying a vector by a scalar (a single number). For example, if you have a vector \(\mathbf{u} = \langle u_1, u_2 \rangle\) and a scalar \(c\), the resulting vector after scalar multiplication is:

• \(c \mathbf{u} = \langle cu_1, cu_2 \rangle\)

Every component of the vector \(\mathbf{u}\) gets multiplied by the scalar \(c\). This operation scales the vector, either stretching or compressing it while maintaining its direction provided \(c\) is positive. If \(c\) is negative, it also reverses the direction of the vector. Scalar multiplication is foundational in many vector calculations and is often used to simplify or transform vectors during problem-solving, such as proving properties involving dot products.

Vector Properties

Vectors are incredibly versatile mathematical objects with distinct properties that differentiate them from other mathematical entities like numbers. Some fundamental properties of vectors include:

• Magnitude: The length or size of the vector, calculated using the Pythagorean theorem for two-dimensional vectors \( \mathbf{u} = \langle u_1, u_2 \rangle\) as \( \sqrt{u_1^2 + u_2^2} \).
• Direction: The orientation of the vector, which remains unchanged during scalar multiplication (excluding the possible direction reversal when the scalar is negative).
• Equality: Two vectors are equal if their corresponding components are equal. For instance, \(\mathbf{u} = \mathbf{v}\) if and only if \(u_1 = v_1\) and \(u_2 = v_2\).
• Addition: Adding two vectors is done component-wise. For two vectors \(\mathbf{u}\) and \(\mathbf{v}\), their sum is \(\langle u_1 + v_1, u_2 + v_2 \rangle\).

Understanding these properties helps in performing various vector operations, such as dot products and scalar multiplications, and is essential for proving vector identities in mathematics.