Problem 75 - Trigonometry

Question

These questions are available for instructors to help assess if you have successfully met the learning objectives for this section. Find the magnitude of vector \(\mathbf{V}=5 \mathbf{i}-2 \mathbf{j}\). a. 10 b. \(\sqrt{29}\) c. 7 d. \(\sqrt{21}\)

Step-by-Step Solution

Answer

The magnitude of vector \( \mathbf{V} \) is \( \sqrt{29} \).

Step 1: Understand the Vector

The vector \( \mathbf{V} = 5 \mathbf{i} - 2 \mathbf{j} \) consists of components 5 in the \( \mathbf{i} \) direction and -2 in the \( \mathbf{j} \) direction. This represents the vector in a two-dimensional space, where \( \mathbf{i} \) and \( \mathbf{j} \) are unit vectors along the x-axis and y-axis, respectively.

Step 2: Apply the Magnitude Formula

The formula for the magnitude of a vector \( \mathbf{V} = a \mathbf{i} + b \mathbf{j} \) is \( \| \mathbf{V} \| = \sqrt{a^2 + b^2} \). Here, \( a = 5 \) and \( b = -2 \). Substitute these values into the formula.

Step 3: Calculate the Magnitude

Substitute the components into the magnitude formula: \[\| \mathbf{V} \| = \sqrt{5^2 + (-2)^2}.\] Calculate the squares: \(5^2 = 25\) and \((-2)^2 = 4\). Add them: \(25 + 4 = 29\). Now, find the square root: \(\| \mathbf{V} \| = \sqrt{29}.\)

Step 4: Compare with Given Options

Look at the answer choices provided: a. 10, b. \(\sqrt{29}\), c. 7, d. \(\sqrt{21}\). The calculated magnitude \(\sqrt{29}\) matches option b.

Key Concepts

Vector Components

When we talk about vector components, imagine breaking down a vector into parts aligned with the coordinate axes. Each vector in a two-dimensional plane can be expressed as a sum of its components in the x and y directions. For example, the vector \( \mathbf{V} = 5 \mathbf{i} - 2 \mathbf{j} \) consists of:

• A component of 5 units along the x-axis (represented by \( \mathbf{i} \) direction), and
• A component of -2 units along the y-axis (represented by \( \mathbf{j} \) direction).

These components tell us how much of the vector points in each direction. This way of representing vectors is crucial because it simplifies calculations such as finding the vector's length, also known as its magnitude.

Unit Vectors

Unit vectors are essential in understanding vectors. They are vectors with a magnitude of exactly one. These little helpers form the basis for more complicated vector expressions. In a two-dimensional space:

• The unit vector \( \mathbf{i} \) points in the positive x-direction,
• The unit vector \( \mathbf{j} \) points in the positive y-direction.

Thus, any vector in this plane, like \( 5 \mathbf{i} - 2 \mathbf{j} \), is essentially a combination of these basic unit vectors, multiplied by how far they stretch in their respective directions. They serve as a framework or skeleton upon which vectors are built.

Magnitude Formula

The magnitude of a vector is like the length of a line that represents the vector from the origin to a certain point in space. To measure this, we use the magnitude formula, which is derived from the Pythagorean theorem. For a vector \( \mathbf{V} = a \mathbf{i} + b \mathbf{j} \), the magnitude \( \| \mathbf{V} \| \) is calculated as: \[ | \mathbf{V} \| = \sqrt{a^2 + b^2} \] For \( \mathbf{V} = 5 \mathbf{i} - 2 \mathbf{j} \), simply substitute \( a = 5 \) and \( b = -2 \). The formula becomes: \[ | \mathbf{V} \| = \sqrt{5^2 + (-2)^2} = \sqrt{29} \] Thus, the magnitude or length of the vector is \( \sqrt{29} \). This gives us a way to "length" a vector using its components.

Two-dimensional Vectors

Vectors that lie in a two-dimensional space have unique characteristics. They are represented by two components, one for each dimension, aligning with the x and y axes. This is like plotting a point on paper where the x-coordinate and y-coordinate determine its position. Using unit vectors \( \mathbf{i} \) and \( \mathbf{j} \), any two-dimensional vector \( \mathbf{V} \) can be written as \( a \mathbf{i} + b \mathbf{j} \).

• The component along \( \mathbf{i} \) indicates movement on the x-axis.
• The component along \( \mathbf{j} \) shows movement on the y-axis.

These vectors are fundamental in physics and engineering for modeling and understanding systems in a planar space. By splitting a vector into components and understanding its magnitude and direction, we gain a comprehensive understanding of how it will behave in that space.