Problem 56 - Algebra and Trigonometry

Question

In Exercises 53–56, let. $$\mathbf{u}=-2 \mathbf{i}+3 \mathbf{j}, \mathbf{v}=6 \mathbf{i}-\mathbf{j}, \mathbf{w}=-3 \mathbf{i}$$ Find each specified vector or scalar. $$|\mathbf{v}+\mathbf{w}|^{2}-\| \mathbf{v}-\left.\mathbf{w}\right|^{2}$$

Step-by-Step Solution

Answer

The result is \(-72\)

Step 1: Calculate \(\mathbf{v}+\mathbf{w}\) and \(\mathbf{v}-\mathbf{w}\)

To calculate the sum and difference of vectors \(\mathbf{v}\) and \(\mathbf{w}\) we simply add/subtract their corresponding components. \(\mathbf{v}+\mathbf{w}=(6\mathbf{i}-\mathbf{j})+(-3\mathbf{i})=3\mathbf{i}-\mathbf{j}\) and \(\mathbf{v}-\mathbf{w}=(6\mathbf{i}-\mathbf{j}) - (-3\mathbf{i})= 9\mathbf{i} - \mathbf{j}\)

Step 2: Calculate the magnitude squared

The magnitude squared of vectors \(\mathbf{v}+\mathbf{w}\) and \(\mathbf{v}-\mathbf{w}\) is obtained by squaring the magnitude. The magnitude of a vector \(\mathbf{a} = x\mathbf{i} + y\mathbf{j}\) is \(\|\mathbf{a}\| = \sqrt{x^2+y^2}\) therefore: \(\|\mathbf{v}+\mathbf{w}\|^2 = (3^2+(-1)^2) = 10\) and \(\|\mathbf{v}-\mathbf{w}\|^2 = (9^2+(-1)^2) = 82\)

Step 3: Compute the difference

Finally, simply subtracting \(\| \mathbf{v}-\mathbf{w}\|^2\) from \(\|\mathbf{v}+\mathbf{w}\|^2\) gives us the result: \(\|\mathbf{v}+\mathbf{w}\|^2 - \| \mathbf{v}-\mathbf{w}\|^2 = 10 - 82 = -72\)

Key Concepts

Vector Addition

Vector addition is a fundamental operation in vector algebra that allows us to combine two or more vectors to produce a resultant vector. To add two vectors, like \(\mathbf{v}\) and \(\mathbf{w}\), you sum their corresponding components. In mathematical terms, if \(\mathbf{v} = a\mathbf{i} + b\mathbf{j}\) and \(\mathbf{w} = c\mathbf{i} + d\mathbf{j}\), then their sum \(\mathbf{v} + \mathbf{w}\) is \((a+c)\mathbf{i} + (b+d)\mathbf{j}\).
This simple addition of components helps in various applications, such as graphical representations or solving physics problems involving directions.
Since vectors are directional entities with both magnitude and direction, vector addition is crucial in fields like engineering and physics for accurately representing combined forces and velocities.

• Notably, vector addition is commutative, meaning \(\mathbf{v} + \mathbf{w} = \mathbf{w} + \mathbf{v}\).

Magnitude of a Vector

The magnitude of a vector represents the length or size of the vector. For a vector expressed in the format \(\mathbf{a} = x\mathbf{i} + y\mathbf{j}\), the magnitude is computed using the Pythagorean theorem, given by the formula \(\|\mathbf{a}\| = \sqrt{x^2 + y^2}\).
This measurement is crucial because it provides the scale for the vector’s direction, making it vital in calculations involving physical quantities like displacement, force, or velocity.
Calculating the magnitude is also essential when comparing different vectors to understand how one vector's size relates to another. For instance, in our problem, after calculating the magnitudes of \(\mathbf{v} + \mathbf{w}\) and subtracting \(\mathbf{v} - \mathbf{w}\), we're handling these magnitudes to find differences in scales.

Scalar Multiplication

Scalar multiplication involves multiplying a vector by a real number (a scalar). This operation changes the magnitude of the vector while preserving its direction if the scalar is positive.
For a vector \(\mathbf{a} = x\mathbf{i} + y\mathbf{j}\) and a scalar \(k\), the product \(k\mathbf{a}\) is \((kx)\mathbf{i} + (ky)\mathbf{j}\).

• If \(k\) is positive, the vector stretches, maintaining the same direction.
• If \(k\) is negative, it flips the vector's direction.

Scalar multiplication is essential for mathematical modeling in physics and engineering by allowing vectors to be resized depending on the specific conditions or constraints of the problem.
It is also used in scaling vectors to unit vectors, which helps in finding direction without regard to magnitude.

Vector Subtraction

Vector subtraction is similar to vector addition but involves finding the difference between two vectors. To subtract vector \(\mathbf{w}\) from vector \(\mathbf{v}\), you subtract the components of \(\mathbf{w}\) from the corresponding components of \(\mathbf{v}\).
If \(\mathbf{v} = a\mathbf{i} + b\mathbf{j}\) and \(\mathbf{w} = c\mathbf{i} + d\mathbf{j}\), then their difference \(\mathbf{v} - \mathbf{w}\) is \((a-c)\mathbf{i} + (b-d)\mathbf{j}\).
Subtraction of vectors is useful in determining the relative position or speed of one object with respect to another, which is frequent in physics problems such as velocity analysis.

• Vector subtraction can be visualized geometrically by the tip-to-tail method, where the second vector is reversed and added to the first.

Understanding this concept enhances problem-solving capabilities where directional differences are crucial, such as navigating planes, boats, or even paths in computer graphics.