Problem 44 - Trigonometry

Question

For each pair of vectors, find \(\mathbf{U}+\mathbf{V}, \mathbf{U}-\mathbf{V}\), and \(2 \mathbf{U}-3 \mathbf{V}\). $$\mathbf{U}=\langle-6,-3\rangle, \mathbf{V}=\langle-2,5\rangle$$

Step-by-Step Solution

Answer

\( \mathbf{U} + \mathbf{V} = \langle -8, 2 \rangle \), \( \mathbf{U} - \mathbf{V} = \langle -4, -8 \rangle \), and \( 2\mathbf{U} - 3\mathbf{V} = \langle -6, -21 \rangle \).

Step 1: Find \( \mathbf{U} + \mathbf{V} \)

To find the sum of the vectors \( \mathbf{U} \) and \( \mathbf{V} \), add the corresponding components. So, for \( \mathbf{U} = \langle -6, -3 \rangle \) and \( \mathbf{V} = \langle -2, 5 \rangle \): \[ \mathbf{U} + \mathbf{V} = \langle -6 + (-2), -3 + 5 \rangle \] Calculate: \[ \mathbf{U} + \mathbf{V} = \langle -8, 2 \rangle \]

Step 2: Find \( \mathbf{U} - \mathbf{V} \)

To find the difference of the vectors \( \mathbf{U} \) and \( \mathbf{V} \), subtract the corresponding components. So, for \( \mathbf{U} = \langle -6, -3 \rangle \) and \( \mathbf{V} = \langle -2, 5 \rangle \): \[ \mathbf{U} - \mathbf{V} = \langle -6 - (-2), -3 - 5 \rangle \] Calculate: \[ \mathbf{U} - \mathbf{V} = \langle -4, -8 \rangle \]

Step 3: Find \( 2\mathbf{U} - 3\mathbf{V} \)

To find the expression \( 2\mathbf{U} - 3\mathbf{V} \), first calculate \( 2\mathbf{U} \) and \( 3\mathbf{V} \), then subtract the results. \( 2\mathbf{U} = 2 \times \langle -6, -3 \rangle = \langle -12, -6 \rangle \)\( 3\mathbf{V} = 3 \times \langle -2, 5 \rangle = \langle -6, 15 \rangle \)Now subtract: \[ 2\mathbf{U} - 3\mathbf{V} = \langle -12 - (-6), -6 - 15 \rangle \] Calculate: \[ 2\mathbf{U} - 3\mathbf{V} = \langle -6, -21 \rangle \]

Key Concepts

Understanding Vectors

Vectors are essential in mathematics and physics. They are quantities that have both magnitude (size) and direction. Unlike regular numbers, which only indicate how much or how many, vectors tell us how fast something is moving and in what direction. In mathematics, we often represent vectors as arrows on a graph, and they work differently from regular numbers when adding, subtracting, or multiplying.

• Vectors in two dimensions can be represented as ordered pairs, such as \( \mathbf{U} = \langle -6, -3 \rangle \).
• The first number (or component) of the pair represents the horizontal direction, and the second represents the vertical direction.

Understanding these basics makes it easier to work with vectors in calculations, as demonstrated in the exercise above.

Exploring Linear Combinations

A linear combination is essentially a way to mix and match vectors to achieve a desired result. It involves multiplying vectors by scalars (real numbers) and then adding the results.For example, when calculating \( 2\mathbf{U} - 3\mathbf{V} \), we:

• First, multiply vector \( \mathbf{U} \) by 2 to get \( \langle -12, -6 \rangle \).
• Next, multiply vector \( \mathbf{V} \) by 3 to get \( \langle -6, 15 \rangle \).
• Finally, subtract the two results to reach the final vector \( \langle -6, -21 \rangle \).

Linear combinations allow us to adjust vectors in magnitude and direction, forming new vectors that can model different situations or solve problems.

Breaking Down Vector Components

Vector components play a crucial role when dealing with vector operations. By understanding how to manipulate these components, adding, subtracting, or altering vectors becomes a straightforward process.Each vector has both an x-component and a y-component:

• The x-component affects horizontal movement, and
• The y-component affects vertical movement.

When adding vectors \( \mathbf{U} \) and \( \mathbf{V} \), it's like combining these horizontal and vertical influences:- For \( \mathbf{U} + \mathbf{V} \), the equation \( \langle -6 + (-2), -3 + 5 \rangle \) gives us new x and y directions. This breaking down helps simplify complex vector problems into manageable components, making calculations precise and efficient.