Problem 19 - APEX Calculus

Question

Find \(\|\vec{u}\|,\|\vec{v}\|,\|\vec{u}+\vec{v}\|\) and \(\|\vec{u}-\vec{v}\|\) \(\vec{u}=\langle 1,2\rangle, \quad \vec{v}=\langle-3,-6\rangle\)

Step-by-Step Solution

Answer

\(\|\vec{u}\| = \sqrt{5}\), \(\|\vec{v}\| = 3\sqrt{5}\), \(\|\vec{u} + \vec{v}\| = 2\sqrt{5}\), \(\|\vec{u} - \vec{v}\| = 4\sqrt{5}\).

Step 1: Calculate the Magnitude of \(\vec{u}\)

The magnitude of a vector \(\vec{u} = \langle a, b \rangle\) is given by the formula \(\|\vec{u}\| = \sqrt{a^2 + b^2}\). For \(\vec{u} = \langle 1, 2 \rangle\), substitute the values: \(\|\vec{u}\| = \sqrt{1^2 + 2^2} = \sqrt{1 + 4} = \sqrt{5}\).

Step 2: Calculate the Magnitude of \(\vec{v}\)

The magnitude of vector \(\vec{v} = \langle -3, -6 \rangle\) is found using the same formula. Substitute \(a = -3\) and \(b = -6\): \(\|\vec{v}\| = \sqrt{(-3)^2 + (-6)^2} = \sqrt{9 + 36} = \sqrt{45} = 3\sqrt{5}\).

Step 3: Add the Vectors \(\vec{u}\) and \(\vec{v}\)

To add vectors \(\vec{u} = \langle 1, 2 \rangle\) and \(\vec{v} = \langle -3, -6 \rangle\), add corresponding components: \(\vec{u} + \vec{v} = \langle 1 + (-3), 2 + (-6) \rangle = \langle -2, -4 \rangle\).

Step 4: Calculate the Magnitude of \(\vec{u} + \vec{v}\)

Calculate the magnitude of the resultant vector \(\langle -2, -4 \rangle\): \(\|\vec{u} + \vec{v}\| = \sqrt{(-2)^2 + (-4)^2} = \sqrt{4 + 16} = \sqrt{20} = 2\sqrt{5}\).

Step 5: Subtract the Vectors \(\vec{u}\) and \(\vec{v}\)

To subtract vector \(\vec{v}\) from \(\vec{u}\), subtract the corresponding components: \(\vec{u} - \vec{v} = \langle 1 - (-3), 2 - (-6) \rangle = \langle 4, 8 \rangle\).

Step 6: Calculate the Magnitude of \(\vec{u} - \vec{v}\)

Calculate the magnitude of the vector \(\langle 4, 8 \rangle\): \(\|\vec{u} - \vec{v}\| = \sqrt{4^2 + 8^2} = \sqrt{16 + 64} = \sqrt{80} = 4\sqrt{5}\).

Key Concepts

Vector Addition

Vector addition is a fundamental operation in which we join two or more vectors to get a resultant vector. Each vector has a magnitude and direction. In 2D space, like in our example, we represent vectors with components in the x and y directions. To add vectors, you simply add their corresponding components together.

For instance, if we have vectors \( \vec{u} = \langle 1, 2 \rangle \) and \( \vec{v} = \langle -3, -6 \rangle \), the sum is:

• Add the x-components: \(1 + (-3) = -2\)
• Add the y-components: \(2 + (-6) = -4\)

Thus, the sum \( \vec{u} + \vec{v} = \langle -2, -4 \rangle \). This new vector represents combined forces, directions, or movements as indicated by the original vectors.

Vector Subtraction

Just like addition, vector subtraction involves combining vectors, but in a different manner. It consists of taking one vector away from another by subtracting their corresponding components.

When subtracting \( \vec{v} \) from \( \vec{u} \), we perform the following operations:

• Subtract the x-components: \(1 - (-3) = 4\)
• Subtract the y-components: \(2 - (-6) = 8\)

So, \( \vec{u} - \vec{v} = \langle 4, 8 \rangle \). Vector subtraction can be thought of as adding the negative of the vector, providing the difference in direction or change in position when compared to the initial state.

Euclidean Norm

The Euclidean norm (or magnitude) of a vector provides a measure of its length or size. For a vector \( \vec{a} = \langle a_1, a_2 \rangle \), its magnitude is calculated using the formula:\[\|\vec{a}\| = \sqrt{a_1^2 + a_2^2}\]
This formula is grounded in the Pythagorean theorem, viewing the vector as the hypotenuse of a right triangle.

In our example:

• For \( \vec{u} = \langle 1, 2 \rangle \), \( \|\vec{u}\| = \sqrt{1^2 + 2^2} = \sqrt{5} \)
• For \( \vec{v} = \langle -3, -6 \rangle \), \( \|\vec{v}\| = \sqrt{9 + 36} = 3\sqrt{5} \)
• For \( \vec{u} + \vec{v} = \langle -2, -4 \rangle \), \( \|\vec{u} + \vec{v}\| = \sqrt{20} = 2\sqrt{5} \)
• For \( \vec{u} - \vec{v} = \langle 4, 8 \rangle \), \( \|\vec{u} - \vec{v}\| = \sqrt{80} = 4\sqrt{5} \)

The Euclidean norm gives insight into how stretched a vector is when it is viewed as a straight line segment.

Vector Operations

Vector operations, which include addition, subtraction, and calculation of magnitudes, are crucial for analyzing physical phenomena in physics, engineering, and mathematics. These operations allow us to manipulate vectors to determine resultant forces, directions, or velocities among others.

When combining vectors, both the magnitude and direction are important:

• **Addition**: Combines directions and magnitudes.
• **Subtraction**: Presents the change or difference between two vectors.
• **Magnitude**: Gives the length, helping understand the vector's strength or speed.

Understanding these operations enables us to simplify complex systems into more manageable components, providing a clear visual representation of vector relationships.