Problem 33 - Precalculus : Building Concepts and Connections

Question

Perform each operation, given \(\mathbf{u}, \mathbf{v},\) and \(\mathbf{w}\) $$\mathbf{u}=\langle 3,2\rangle, \mathbf{v}=\langle-1,4\rangle, \mathbf{w}=\langle-2,-1\rangle$$ $$2 u+3 w$$

Step-by-Step Solution

Answer

The answer to the vector operation \(2\mathbf{u} + 3\mathbf{w}\) is \(\langle0,1\rangle\).

Step 1: Scalar Multiplication of Vector \(\mathbf{u}\)

To perform scalar multiplication of a vector \(\mathbf{u}=\langle 3,2\rangle\) by 2, multiply 2 with each of the member of vector \(\mathbf{u}\). This gives: \(2\mathbf{u} = 2 \times\langle 3,2\rangle = \langle 2 \times 3, 2 \times 2\rangle = \langle6,4\rangle\).

Step 2: Scalar Multiplication of Vector \(\mathbf{w}\)

To perform scalar multiplication of a vector \(\mathbf{w}=\langle-2,-1\rangle\) by 3, multiply 3 with each of the member of vector \(\mathbf{w}\). This gives: \(3\mathbf{w} = 3\times\langle-2,-1\rangle = \langle 3 \times -2, 3 \times -1\rangle = \langle-6,-3\rangle\).

Step 3: Vector Addition

To add the vectors from Steps 1 and 2, you sum the corresponding entries. This gives: \(2\mathbf{u} +3\mathbf{w}=\langle6,4\rangle+\langle-6,-3\rangle = \langle 6 + (-6), 4 + (-3)\rangle = \langle0,1\rangle\).

Key Concepts

Scalar Multiplication of Vectors

Scalar multiplication of a vector is a fundamental operation in vector algebra, and it has a variety of applications in fields like physics, engineering, and computer science. It involves multiplying a vector by a scalar, which is simply a real number. In this process, each component of the vector is multiplied by the scalar, resulting in a new vector.

For example, consider a vector \( \mathbf{u} = \langle 3,2 \rangle \). To perform scalar multiplication of this vector by 2, you would calculate \( 2\mathbf{u} = 2 \times \langle 3,2 \rangle = \langle 6,4 \rangle \). This new vector points in the same (or opposite, if the scalar is negative) direction as the original but is scaled by the factor of 2, which means its length is twice as long.

Students often confuse scalar multiplication with dot product, another vector operation, but it is important to remember that scalar multiplication results in a vector, while dot product results in a scalar.

Visual Representation of Scalar Multiplication

A good tip to visualize scalar multiplication is to sketch the vector and then alter its length according to the scalar. If you scale vector \( \mathbf{u} \) by 2, you'll simply stretch it to twice its original length in the same direction.

Vector Addition

Vector addition is the process of combining two or more vectors into one single vector called the resultant. This operation adheres to the 'tip-to-tail' method or parallelogram rule. During this process, the vectors retain their individual magnitudes and directions.

Let's now apply this to our example vectors \( 2\mathbf{u} = \langle 6,4 \rangle \) and \( 3\mathbf{w} = \langle -6,-3 \rangle \). When added together, you sum up the corresponding components to find the resultant vector. So, \( 2\mathbf{u} + 3\mathbf{w} = \langle 6 + (-6), 4 + (-3) \rangle = \langle 0,1 \rangle \). The resulting vector is a new entity that characterizes the cumulative effect of both vectors involved.

Remember, vector addition is commutative; the order in which you add the vectors does not affect the result.

Common Mistakes in Vector Addition

A common mistake when learning vector addition is to mix up the components, but it's crucial to match the corresponding components correctly to ensure a correct resultant vector. Also, do not confuse this operation with adding magnitudes; vectors must be added component-wise.

Precalculus Vector Problems

Vector problems in precalculus set the stage for more complex vector applications in calculus and other advanced mathematics. To master precalculus vector problems, it's crucial to have a solid understanding of vector operations, as they're often the building blocks used to unravel more complicated scenarios, such as those involving speed, velocity, and forces.

When working with vectors in precalculus, you'll commonly deal with exercises that involve vector addition, scalar multiplication, dot product, and cross product. To effectively tackle these problems, develop a methodical approach: identify known vectors, interpret the problem to determine which operations to apply, and execute the operations carefully to avoid simple arithmetic errors.

Visualization Techniques

Utilizing a graphical approach can greatly assist in solving precalculus vector problems. Sketching vectors and their resultant can make abstract problems more tangible and conceptually easier to solve. Whether it's plotting vectors on a graph or imagining them in a physical context, visualizing these concepts is an invaluable tool for understanding their principles and applications.