Problem 61 - Precalculus

Question

Find the dot product \langle-3,2\rangle\(\cdot\langle 2,5\rangle\) Solution: Multiply component by component. \(\langle-3,2\rangle \cdot\langle 2,5\rangle=\langle(-3)(2),(2)\rangle\) Simplify. \(\langle-3,2\rangle \cdot\langle 2,5\rangle=\langle-6,10\rangle\) This is incorrect. What mistake was made?

Step-by-Step Solution

Answer

The mistake was in calculation of the dot product components. The correct dot product is 4.

Step 1: Understand the Dot Product

The dot product (also known as the scalar product) of two vectors \(\langle a, b \rangle\) and \(\langle c, d \rangle\) is calculated as follows:\[ a \cdot c + b \cdot d \]

Step 2: Apply the Dot Product Formula

Apply the dot product formula to the vectors \(-3, 2\) and \(2, 5\) as follows:\[-3 \cdot 2 + 2 \cdot 5 \]

Step 3: Calculate Each Multiplication

Calculate each multiplication individually:- \(-3 \times 2 = -6\)- \(2 \times 5 = 10\)

Step 4: Add the Results

Add the two results from Step 3:\[-6 + 10\]

Step 5: Final Calculation

Calculate the sum:\[-6 + 10 = 4\]

Key Concepts

Dot Product

The Dot Product, also known as the Scalar Product, is a fundamental operation in vector calculus. It combines two vectors to return a single number, or scalar. This operation is useful in many fields, including physics and computer graphics. The formula for the dot product of two vectors \(\langle a, b \rangle\) and \(\langle c, d \rangle\) is given by:

• \(a \cdot c + b \cdot d\)

To understand the dot product, it helps to think about projecting one vector onto another. The result is a measure of how much of one vector goes in the direction of the other. This is crucial in problems involving angles between vectors. Remember, vectors must be of the same dimension for the dot product to be defined.
For example, given vectors \(\langle -3, 2 \rangle\) and \(\langle 2, 5 \rangle\), the dot product is calculated by multiplying corresponding components and summing the results. This leads to the scalar value \(-6 + 10 = 4\). The dot product can help determine whether two vectors are perpendicular. If the result is zero, the vectors are orthogonal.

Scalar Product

The Scalar Product is another name for the Dot Product. It results in a scalar rather than a vector, which means it only has magnitude and no direction. The scalar product is essential in understanding the relationship between two vectors.

• Think "dot" in terms of multiplication, as it combines elements multiplicatively.
• A scalar result simplifies complex vector interactions to a single value.

A key point is that the scalar product respects the properties of commutativity and distributivity. For the same pair of vectors, regardless of order, the scalar product of vectors \(\langle a, b \rangle\) and \(\langle c, d \rangle\), as \(a \cdot c + b \cdot d\), will give the same result.
This property makes the scalar product very handy in simplifying vector equations and computations in mathematical physics and engineering contexts.

Vectors

Vectors are mathematical objects used to represent quantities that have both magnitude and direction. They are often depicted as arrows in a coordinate system. In a 2D coordinate plane, vectors are represented as \(\langle x, y \rangle\). Vectors are useful in a wide range of applications, from physics to engineering to computer graphics.

• The algebra of vectors involves operations such as addition, subtraction, and scalar multiplication.
• Vectors can be decomposed into components to simplify analysis in complex systems.

In the exercise, the vectors \(\langle-3, 2\rangle\) and \(\langle2, 5\rangle\) were the focus. These vectors can represent any quantity that has both direction and magnitude, such as displacement, velocity, or force. Analyzing them via dot product gives insights into their directional relationship. Understanding vector operations forms the foundation for many advanced mathematical techniques in calculus and linear algebra.

Precalculus Concepts

Precalculus is an essential building block for advanced mathematics. It introduces concepts that prepare students for calculus, such as functions, series, and importantly, vectors and their operations.

• Fundamental understanding of different types of numbers and operations sets the stage for calculus.
• Mastery of vector operations, like the dot product, is crucial for solving real-world problems.

The dot product, calculated as \(-3 \times 2 + 2 \times 5 = 4\), is a perfect example of precalculus algebra in action. It requires knowledge of basic arithmetic and understanding of vector geometry.
These concepts are not only fundamental in mathematics but also applicable in computing, physics, and engineering scenarios daily. Mastery of these ideas will ensure students are well-prepared for the challenges of calculus and beyond.