Problem 17 - APEX Calculus

Question

In Exercises 17-20, find \(\|\vec{u}\|,\|\vec{v}\|,\|\vec{u}+\vec{v}\|\) and \(\|\vec{u}-\vec{v}\|\) \(\vec{u}=\langle 2,1\rangle, \quad \vec{v}=\langle 3,-2\rangle\)

Step-by-Step Solution

Answer

\(\|\vec{u}\| = \sqrt{5}\), \(\|\vec{v}\| = \sqrt{13}\), \(\|\vec{u} + \vec{v}\| = \sqrt{26}\), \(\|\vec{u} - \vec{v}\| = \sqrt{10}\).

Step 1: Magnitude of Vector u

To find the magnitude of vector \( \vec{u} = \langle 2, 1 \rangle \), use the formula \( \| \vec{u} \| = \sqrt{x^2 + y^2} \). Calculate:\[\|\vec{u}\| = \sqrt{2^2 + 1^2} = \sqrt{4 + 1} = \sqrt{5}\]

Step 2: Magnitude of Vector v

For the magnitude of vector \( \vec{v} = \langle 3, -2 \rangle \), apply the same formula:\[\|\vec{v}\| = \sqrt{3^2 + (-2)^2} = \sqrt{9 + 4} = \sqrt{13}\]

Step 3: Sum of Vectors u and v

Add vectors \( \vec{u} \) and \( \vec{v} \): \[\vec{u} + \vec{v} = \langle 2, 1 \rangle + \langle 3, -2 \rangle = \langle 2+3, 1-2 \rangle = \langle 5, -1 \rangle\]

Step 4: Magnitude of u + v

Find the magnitude of \( \vec{u} + \vec{v} = \langle 5, -1 \rangle \):\[\|\vec{u} + \vec{v}\| = \sqrt{5^2 + (-1)^2} = \sqrt{25 + 1} = \sqrt{26}\]

Step 5: Difference of Vectors u and v

Subtract vector \( \vec{v} \) from \( \vec{u} \): \[\vec{u} - \vec{v} = \langle 2, 1 \rangle - \langle 3, -2 \rangle = \langle 2-3, 1+2 \rangle = \langle -1, 3 \rangle\]

Step 6: Magnitude of u - v

Find the magnitude of \( \vec{u} - \vec{v} = \langle -1, 3 \rangle \):\[\|\vec{u} - \vec{v}\| = \sqrt{(-1)^2 + 3^2} = \sqrt{1 + 9} = \sqrt{10}\]

Key Concepts

Understanding the Magnitude of a Vector

The magnitude of a vector is essentially its length or size, represented by the symbol \( \|\vec{u}\| \). It is derived using the Pythagorean theorem in a coordinate system. To compute it for a vector \( \vec{u} = \langle x, y \rangle \), we use the formula:

• \( \|\vec{u}\| = \sqrt{x^2 + y^2} \)

This formula gives us the distance of the vector from the origin \( \langle 0, 0 \rangle \) to the point \( \langle x, y \rangle \) on the Cartesian plane.
For instance, for vector \( \vec{u} = \langle 2, 1 \rangle \), its magnitude is \( \sqrt{5} \), meaning it stretches \( \sqrt{5} \) units from the origin. It's important to note that magnitude is always a non-negative number.

Exploring Vector Addition

Vector addition involves combining two or more vectors to form a new vector. This operation is crucial when we want to derive a resultant vector from multiple ones. When adding vectors, you add their corresponding components:

• \( \vec{u} = \langle u_1, u_2 \rangle \)
• \( \vec{v} = \langle v_1, v_2 \rangle \)

The sum \( \vec{u} + \vec{v} \) is calculated as \( \langle u_1 + v_1, u_2 + v_2 \rangle \).
For example, adding vectors \( \vec{u} = \langle 2, 1 \rangle \) and \( \vec{v} = \langle 3, -2 \rangle \) results in the vector \( \langle 5, -1 \rangle \).
This means the resultant vector moves 5 units directly along the x-axis and 1 unit downwards on the y-axis from the origin.

Decoding Vector Subtraction

Vector subtraction computes the difference between two vectors, essentially reversing the direction of the subtracted vector and then adding it to the other. The operation is performed component-wise as follows:

• \( \vec{u} - \vec{v} = \langle u_1 - v_1, u_2 - v_2 \rangle \)

Subtracting vector \( \vec{v} \) from \( \vec{u} \) gives us \( \langle -1, 3 \rangle \).
In this case, \( \vec{u} = \langle 2, 1 \rangle \) reduces by vector \( \vec{v} = \langle 3, -2 \rangle \).
Think of vector subtraction as the inverse of addition; it's like removing the effect of one vector from another, resulting in a new vector.

Vector Concepts in Calculus Exercises

Vectors play a significant role in calculus, appearing in various problems that deal with rates of change and motion. Understanding operations like magnitude and addition is crucial in calculus exercises. For example:

• Magnitude becomes important in determining the speed of an object.
• Vector addition might be used to analyze forces in physics problems.

When dealing with calculus problems that employ vectors, you start by defining vectors and computing their key characteristics.
These exercises often help in visualizing complex movements or changes over time through vector drawings.
Such exercises are especially beneficial for understanding multidimensional changes in a system, illustrating how vectors directly influence calculus' higher-level operations.