Problem 20 - Precalculus with Limits

Question

In Exercises 15-24, use the vectors \(\mathbf{u} = \langle 3, 3 \rangle\), \(\mathbf{v} = \langle -4, 2 \rangle\), and \(\mathbf{w} = \langle 3, -1 \rangle\) to find the indicated quantity. State whether the result is a vector or a scalar. \((\mathbf{u} \cdot 2\mathbf{v}) \mathbf{w}\)

Step-by-Step Solution

Answer

The result \((\mathbf{u} \cdot 2\mathbf{v}) \mathbf{w} = \langle -36, 12 \rangle\) is a vector.

Step 1: Scalar Multiplication

Multiply \(\mathbf{v} = \langle -4, 2 \rangle\) by 2, resulting in a new vector \(2\mathbf{v} = \langle -8, 4 \rangle\)

Step 2: Dot Product

Compute the dot product between \(\mathbf{u} = \langle 3, 3 \rangle\) and \(2\mathbf{v} = \langle -8, 4 \rangle\). The dot product of two vectors \(\mathbf{u}=\langle u_{1}, u_{2}\rangle\) and \(\mathbf{v}=\langle v_{1}, v_{2}\rangle\) is defined as \(u_{1}*v_{1}+u_{2}*v_{2}\), hence we get \(\mathbf{u} \cdot 2\mathbf{v} = (3*(-8))+(3*4) = -24+12=-12\)

Step 3: Scalar Multiplication

Finally, multiply \(\mathbf{w} = \langle 3, -1 \rangle\) with the scalar value from the dot product in previous step. Therefore, \(-12\mathbf{w} = \langle -36, 12 \rangle\).

Key Concepts

Scalar Multiplication

Scalar multiplication is a fundamental operation in vector arithmetic where a vector is multiplied by a scalar (a real number), resulting in a new vector that is parallel to the original but with a length scaled by the scalar value.

In the context of vector operations in precalculus, scalar multiplication affects each component of a vector equally. For instance, consider a vector \(\mathbf{a} = \langle a_1, a_2 \rangle\) and a scalar \(c\). The result of the scalar multiplication \(c\mathbf{a}\) is given by \(\langle c\cdot a_1, c\cdot a_2 \rangle\).

This property is particularly useful for changing the magnitude of a vector while keeping its direction unchanged. Scalar multiplication also plays a critical role in the resolution of vectors into their components and the calculation of dot products, as it prepares vectors to be combined through addition or subtraction with others.

Dot Product

The dot product, also known as the scalar product, is an operation that takes two vectors and returns a single scalar value. This operation is central to many aspects of vector manipulation and is instrumental in determining the angles between vectors and projecting one vector onto another.

Mathematically, if you have two vectors \(\mathbf{u} = \langle u_1, u_2 \rangle\) and \(\mathbf{v} = \langle v_1, v_2 \rangle\), their dot product is computed as \(\mathbf{u} \cdot \mathbf{v} = u_1 \times v_1 + u_2 \times v_2\). The resulting scalar quantity can reveal whether the vectors are orthogonal (perpendicular) if the dot product is zero, or it can provide the cosine of the angle between them when both vectors are normalized.

In the given exercise, the dot product feature allows the composition of the fundamental vector operations to produce a scalar, which then acts as a stepping stone to further vector transformations such as subsequent scalar multiplications.

Vectors in Precalculus

Vectors in precalculus serve as a bridge between basic algebra and more advanced topics in calculus. They give students a tangible grasp on the concept of both magnitude and direction.

Vectors are depicted as directed line segments and are often described by coordinates in two or three dimensions, such as \(\mathbf{u} = \langle u_1, u_2 \rangle\). Precalculus students learn to perform various operations with vectors, including addition, subtraction, scalar multiplication, and dot products. These operations are foundational for understanding motion and forces in physics, maximizing or minimizing functions in optimization problems, and dealing with multi-dimensional data in various fields of study.

Understanding vector operations is essential for any student venturing into fields involving geometry, physics, engineering, computer science, and more. Mastery of the basics of vector arithmetic prepares students for the challenges of calculus, linear algebra, and beyond.