Problem 120 - Algebra and Trigonometry

Question

Exercises \(119-121\) will help you prepare for the material covered in the next section. Find the obtuse angle \(\theta,\) rounded to the nearest tenth of a degree, satisfying. If \(w=-2 i+6 j,\) find the following vector: $$ \frac{2(-2)+4(-6)}{|\mathbf{w}|^{2}} \mathbf{w} $$

Step-by-Step Solution

Answer

The vector after performing the given operation is \(1.4i - 4.2j\).

Step 1: Calculate the magnitude (length) of the vector

The magnitude of a vector is the length of the vector. It is found by using the formula \(\sqrt{x^2+y^2}\), where \(x\) and \(y\) are the values for \(i\) and \(j\) in the respective vector. Thus, the magnitude for vector \(w=-2i+6j\) would be \(\sqrt{(-2)^2 + 6^2} = \sqrt{40}\). So, \(|w|\) equals \(2\sqrt{10}\). This also means that \(|w|^2 = 40\).

Step 2: Substitute the magnitude and vector values into the given equation

The equation given is \(\frac{2(-2)+4(-6)}{|w|^2}w = \frac{2(-2)+4(-6)}{40}(-2i + 6j)\). Substituting the values into the equation we get: \(\frac{-4-24}{40}(-2i + 6j)= -\frac{7}{10}(-2i + 6j)\).

Step 3: Resolve the multiplication

Multiplying the scalar with the vector will yield: \(1.4i - 4.2j\).

Key Concepts

Vector Algebra

Vector algebra is a fundamental part of physics and engineering, serving as a cornerstone for various calculations involving direction and magnitude. Unlike regular algebra that deals with scalar quantities (which have only magnitude), vector algebra involves vector quantities which have both magnitude and direction.

Key operations in vector algebra include vector addition, vector subtraction, and scalar multiplication. To better understand vector addition, imagine you are walking 5 meters east and then 10 meters north. Your total displacement can be represented as the sum of two vectors - one in the east direction and the other in the north direction. Similarly, if you were to reverse your path, that would be akin to subtracting these vectors. The resultants of these operations are vectors themselves, meaning they also possess direction and magnitude.

Vector algebra is vital for solving complex problems in multiple dimensions. It allows the conversion of geometric problems into algebraic form, which is often simpler to manipulate and solve. By mastering vector operations, students can easily handle tasks involving forces, velocities, and other physical phenomena described by vectors.

Magnitude of a Vector

Understanding the magnitude of a vector is critical in grasping the basics of vector algebra. The magnitude, or length, quantifies how far the vector would reach if it were to start at a given point and is a measure of its size. It is a scalar quantity, which means it is described by a numerical value without a direction.

For a two-dimensional vector given by \(\mathbf{v} = x\mathbf{i} + y\mathbf{j}\), where \(\mathbf{i}\) and \(\mathbf{j}\) are the unit vectors in the direction of the x-axis and y-axis, respectively, the magnitude is found using the formula \( |\mathbf{v}| = \sqrt{x^2 + y^2} \). This formula is a direct application of the Pythagorean theorem from right-angled triangles, considering the x and y components as sides of a right-angled triangle.

In the context of the given exercise, the magnitude of vector \(\mathbf{w}\) helps us to further understand its properties and provides a foundation for subsequent operations like scalar multiplication. The calculation of the magnitude is an essential step, which often simplifies the vector equations and makes further vector operations feasible.

Scalar Multiplication of Vectors

Scalar multiplication of vectors involves multiplying a vector by a scalar (a real number), which changes the magnitude of the vector without altering its direction. In essence, scalar multiplication can stretch or shrink a vector, and even reverse its direction if the scalar is negative.

For a vector \(\mathbf{v}\) and scalar \(k\), the scalar multiplication is written as \(k\mathbf{v}\). If \(\mathbf{v} = x\mathbf{i} + y\mathbf{j}\), then \(k\mathbf{v} = kx\mathbf{i} + ky\mathbf{j}\). This operation is applied to each component of the vector, resulting in a new vector that has been scaled accordingly.

In our particular exercise, after calculating the magnitude of vector \(\mathbf{w}\), scalar multiplication is used to solve the given formula. The process of scalar multiplication clarifies the impact of scalar quantities on vector direction and magnitude, emphasizing the link between algebraic expressions and geometric interpretations within the realm of vector algebra.