Problem 2 - College Trigonometry

Question

Use the given pair of vectors \(\vec{v}\) and \(\vec{w}\) to find the following quantities. State whether the result is a vector or a scalar. $$ \begin{array}{lllll} \bullet \vec{v}+\vec{w} & \bullet \vec{w}-2 \vec{v} & \bullet\|\vec{v}+\vec{w}\| & \bullet\|\vec{v}\|+\|\vec{w}\| & \bullet\|\vec{v}\| \vec{w}-\|\vec{w}\| \vec{v} & \bullet\|\vec{w}\| \hat{v} \end{array} $$ Finally, verify that the vectors satisfy the Parallelogram Law $$\|\vec{v}\|^{2}+\|\vec{w}\|^{2}=\frac{1}{2}\left[\|\vec{v}+\vec{w}\|^{2}+\|\vec{v}-\vec{w}\|^{2}\right]$$ $$ \vec{v}=\langle-7,24\rangle, \vec{w}=\langle-5,-12\rangle $$

Step-by-Step Solution

Answer

Vector results: \(\vec{v}+\vec{w}, \vec{w}-2\vec{v}, \|\vec{v}\|\vec{w}-\|\vec{w}\|\vec{v}, \|\vec{w}\|\hat{v}\). Scalar results: \(\|\vec{v}+\vec{w}\|, \|\vec{v}\|+\|\vec{w}\|\). Parallelogram Law is verified.

Step 1: Addition of Vectors

Calculate \(\vec{v}+\vec{w}\) by adding the corresponding components of each vector: \(\vec{v} = \langle-7, 24\rangle\) and \(\vec{w} = \langle-5, -12\rangle\). \[\vec{v}+\vec{w} = \langle -7 + (-5), 24 + (-12) \rangle = \langle -12, 12 \rangle\]This result is a vector.

Step 2: Scalar Multiplication and Subtraction

Calculate \(\vec{w}-2\vec{v}\). First, multiply \(\vec{v}\) by 2: \[2\vec{v} = 2 \times \langle -7, 24 \rangle = \langle -14, 48 \rangle\]Then subtract from \(\vec{w}\):\[\vec{w} - 2\vec{v} = \langle -5, -12 \rangle - \langle -14, 48 \rangle = \langle 9, -60 \rangle\]This result is a vector.

Step 3: Magnitude of Vector Sum

Calculate \(\|\vec{v}+\vec{w}\|\) using the prior result: \(\langle -12, 12 \rangle\).\[\|\vec{v}+\vec{w}\| = \sqrt{(-12)^2 + 12^2} = \sqrt{144 + 144} = \sqrt{288} = 12\sqrt{2}\]This result is a scalar.

Step 4: Sum of Magnitudes

Calculate \(\|\vec{v}\| + \|\vec{w}\|\). First find \(\|\vec{v}\|\) and \(\|\vec{w}\|\):\[\|\vec{v}\| = \sqrt{(-7)^2 + 24^2} = \sqrt{49 + 576} = 25\]\[\|\vec{w}\| = \sqrt{(-5)^2 + (-12)^2} = \sqrt{25 + 144} = 13\]Thus, \(\|\vec{v}\| + \|\vec{w}\| = 25 + 13 = 38\).This result is a scalar.

Step 5: Combination of Magnitudes and Vectors

Calculate \(\|\vec{v}\| \vec{w} - \|\vec{w}\| \vec{v}\):Multiply \(\vec{w}\) by \(\|\vec{v}\|\):\[\|\vec{v}\| \vec{w} = 25 \times \langle -5, -12 \rangle = \langle -125, -300 \rangle\]Multiply \(\vec{v}\) by \(\|\vec{w}\|\):\[\|\vec{w}\| \vec{v} = 13 \times \langle -7, 24 \rangle = \langle -91, 312 \rangle\]Subtract these results:\[\|\vec{v}\| \vec{w} - \|\vec{w}\| \vec{v} = \langle -125, -300 \rangle - \langle -91, 312 \rangle = \langle -34, -612 \rangle\]This result is a vector.

Step 6: Direction Vector Magnitude

Calculate \(\|\vec{w}\| \hat{v}\), where \(\hat{v} = \frac{\vec{v}}{\|\vec{v}\|}\):\[\hat{v} = \frac{1}{25} \langle -7, 24 \rangle = \langle -0.28, 0.96 \rangle\]Multiply by \(\|\vec{w}\|\):\[\|\vec{w}\| \hat{v} = 13 \times \langle -0.28, 0.96 \rangle = \langle -3.64, 12.48 \rangle\]This result is a vector.

Step 7: Verify Parallelogram Law

Verify if \(\|\vec{v}\|^{2}+\|\vec{w}\|^{2}=\frac{1}{2}\left[\|\vec{v}+\vec{w}\|^{2}+\|\vec{v}-\vec{w}\|^{2}\right]\):Calculate \(\|\vec{v}+\vec{w}\|^2\): \[\|\vec{v}+\vec{w}\|^2 = (12\sqrt{2})^2 = 288\]Calculate \(\vec{v}-\vec{w}\): \[\vec{v}-\vec{w} = \langle -7, 24 \rangle - \langle -5, -12 \rangle = \langle -2, 36 \rangle\]Calculate its magnitude: \[\|\vec{v}-\vec{w}\| = \sqrt{(-2)^2 + 36^2} = \sqrt{4 + 1296} = \sqrt{1300} = 10\sqrt{13}\]Calculate \(\|\vec{v}-\vec{w}\|^2\): \[\|\vec{v}-\vec{w}\|^2 = (10\sqrt{13})^2 = 1300\]Verify:\[\|\vec{v}\|^{2}+\|\vec{w}\|^{2} = 25^2 + 13^2 = 625 + 169 = 794\]\[\frac{1}{2}(\|\vec{v}+\vec{w}\|^2 + \|\vec{v}-\vec{w}\|^2) = \frac{1}{2}(288 + 1300) = \frac{1}{2}(1588) = 794\]The Parallelogram Law holds.

Key Concepts

Vector Addition

When working with vectors, one of the fundamental operations you will encounter is vector addition. This involves combining two or more vectors to form a new vector. In the context of this task, we are given two vectors \(\vec{v} = \langle -7, 24 \rangle\) and \(\vec{w} = \langle -5, -12 \rangle\). To add these vectors, simply add their corresponding components: the first component of \(\vec{v}\) with the first component of \(\vec{w}\), and the same for their second components.

• Sum of first components: \(-7 + (-5) = -12\).
• Sum of the second components: \(24 + (-12) = 12\).

Therefore, the result of \(\vec{v} + \vec{w}\) is the vector \(\langle -12, 12 \rangle\). Vector addition is a straightforward but powerful tool, essential for solving more complex problems.

Vector Subtraction

Vector subtraction is another key operation you will use when dealing with vectors. It involves subtracting one vector from another, which can be thought of as adding a negative vector. In performing the operation \(\vec{w} - 2\vec{v}\), first you need to scale \(\vec{v}\) by 2:

• Multiply \(\vec{v} = \langle -7, 24 \rangle\) by 2 to get \(\langle -14, 48 \rangle\).

Next, perform the subtraction of the scaled \(\vec{v}\) from \(\vec{w}\):

• First component: \(-5 - (-14) = 9\)
• Second component: \(-12 - 48 = -60\)

The result of \(\vec{w} - 2\vec{v}\) is \(\langle 9, -60 \rangle\). This outcome remains a vector, demonstrating the principle that vector subtraction effectively reverses direction, aligning it with subtraction of scalars but in a multidimensional context.

Magnitude of a Vector

The magnitude of a vector, often conceptualized as its "length" or "size," is another vital concept. It measures how long or intense the vector is, irrespective of its direction. To determine the magnitude of a vector \(\vec{a} = \langle a_1, a_2 \rangle\), use the formula:\[\|\vec{a}\| = \sqrt{a_1^2 + a_2^2}\]Let's apply it to the vectors \(\vec{v}\) and \(\vec{w}\):

• Magnitude of \(\vec{v}\): \(\sqrt{(-7)^2 + 24^2} = \sqrt{49 + 576} = 25\).
• Magnitude of \(\vec{w}\): \(\sqrt{(-5)^2 + (-12)^2} = \sqrt{25 + 144} = 13\).

Notice that the magnitude is a scalar, representing the overall size of the vector, and is always non-negative. This concept is critical for calculations involving distances in vector spaces.

Parallelogram Law

The Parallelogram Law is a geometrical theorem that applies to vectors, especially in physics and engineering. It states that the sum of the squares of the lengths of the two sides of a parallelogram equals the sum of the squares of the diagonals. More formally, for vectors \(\vec{v}\) and \(\vec{w}\), the law can be expressed as:\[\|\vec{v}\|^2 + \|\vec{w}\|^2 = \frac{1}{2} \left[\|\vec{v} + \vec{w}\|^2 + \|\vec{v} - \vec{w}\|^2\right]\]In our exercise, you verify this law holds by computing:

• \(\|\vec{v} + \vec{w}\| = 12\sqrt{2}\), thus \(\|\vec{v} + \vec{w}\|^2 = 288\).
• \(\|\vec{v} - \vec{w}\| = 10\sqrt{13}\), therefore \(\|\vec{v} - \vec{w}\|^2 = 1300\).

Adding these, halved, indeed equals \(\|\vec{v}\|^2 + \|\vec{w}\|^2\). This confirms the law, illustrating the predictable and structured relationship between vector magnitudes.