Problem 19 - 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. \((3\mathbf{w} \cdot \mathbf{v}) \mathbf{u}\)

Step-by-Step Solution

Answer

The result is a vector \(\langle -126, -126 \rangle\)

Step 1: Calculate the dot product

Calculate the dot product of vectors \(\mathbf{w}\) and \(\mathbf{v}\). With \(\mathbf{w} = \langle 3, -1 \rangle\) and \(\mathbf{v} = \langle -4, 2 \rangle\), you can do this by multiplying each corresponding pair of components from the two vectors and adding the results together. This results in \(3*-4 + -1*2 = -12 - 2 = -14\).

Step 2: Multiply the dot product by 3

Now multiply the result from step 1 by 3 (as described by the exercise). This gives you \(3*-14 = -42\).

Step 3: Multiply the vector \(\mathbf{u}\) by the scalar

Finally you multiply every component of the vector \(\mathbf{u} = \langle 3, 3 \rangle\) by the scalar -42 (result from step 2). Therefore, the resulting vector is \(\langle-126, -126\rangle\).

Key Concepts

Understanding the Dot Product

The dot product is a fundamental operation for vectors, helping us determine the magnitude of a product between two vectors. When we take the dot product of two vectors, we essentially find the sum of the products of their respective components. In this exercise, you used vectors \( \mathbf{w} = \langle 3, -1 \rangle \) and \( \mathbf{v} = \langle -4, 2 \rangle \). By multiplying respective components and then adding them, our equation becomes:

• \(3 \times -4 = -12\)
• \(-1 \times 2 = -2\)

After adding these results, \(-12 + (-2)\), the dot product was found to be \(-14\). Remember, the result of a dot product is always a scalar (a single number), not a vector.

What is Scalar Multiplication?

Scalar multiplication involves taking a scalar, which is just a regular number, and multiplying it by a vector. This operation scales the entire vector by that number without changing its direction, unless the scalar is negative. In the solution, step 2 required multiplying the scalar 3 by the previously obtained dot product result of \(-14\). This resulted in a new scalar, \(-42\). In essence, the dot product, a scalar, was scaled by 3, yielding another scalar.

This process is useful when you need to change the magnitude of a vector or a product of vectors in vector space.

Exploring Vector Multiplication

Vector multiplication can involve different types of operations, including multiplying a vector by another vector (like the dot product) or multiplying a vector by a scalar. In this problem's final step, vector multiplication was performed by multiplying each component of vector \( \mathbf{u} = \langle 3, 3 \rangle \) by the scalar \(-42\) obtained in the previous step. This is how we calculated

• \(3 \times -42 = -126\)
• Again, \(3 \times -42 = -126\)

Thus, the resulting vector became \( \langle -126, -126 \rangle \). This confirms how multiplying a vector by a scalar impacts each component, resulting in a new vector with altered magnitude.