Problem 22 - Algebra and Trigonometry

Question

Find \(2 \mathbf{u},-3 \mathbf{v}, \mathbf{u}+\mathbf{v},\) and \(3 \mathbf{u}-4 \mathbf{v}\) for the given vectors \(\mathbf{u}\) and \(\mathbf{v} .\) $$ \mathbf{u}=\mathbf{i}+\mathbf{j}, \quad \mathbf{v}=\mathbf{i}-\mathbf{j} $$

Step-by-Step Solution

Answer

2\mathbf{u} = 2\mathbf{i} + 2\mathbf{j}, -3\mathbf{v} = -3\mathbf{i} + 3\mathbf{j}, \mathbf{u} + \mathbf{v} = 2\mathbf{i}, 3\mathbf{u} - 4\mathbf{v} = -\mathbf{i} + 7\mathbf{j}.

Step 1: Calculate the Scalar Multiple of Vector \(\mathbf{u}\)

To find \( 2\mathbf{u} \), multiply each component of \( \mathbf{u} = \mathbf{i} + \mathbf{j} \) by 2: \( 2\mathbf{u} = 2(\mathbf{i} + \mathbf{j}) = 2\mathbf{i} + 2\mathbf{j} \).

Step 2: Calculate the Scalar Multiple of Vector \(\mathbf{v}\)

To find \( -3\mathbf{v} \), multiply each component of \( \mathbf{v} = \mathbf{i} - \mathbf{j} \) by -3: \( -3\mathbf{v} = -3(\mathbf{i} - \mathbf{j}) = -3\mathbf{i} + 3\mathbf{j} \).

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

To find \( \mathbf{u} + \mathbf{v} \), add the corresponding components of \( \mathbf{u} = \mathbf{i} + \mathbf{j} \) and \( \mathbf{v} = \mathbf{i} - \mathbf{j} \): \( \mathbf{u} + \mathbf{v} = (\mathbf{i} + \mathbf{j}) + (\mathbf{i} - \mathbf{j}) = 2\mathbf{i} + 0\mathbf{j} = 2\mathbf{i} \).

Step 4: Compute the Combination \(3\mathbf{u} - 4\mathbf{v}\)

First, calculate \( 3\mathbf{u} \) by multiplying \( \mathbf{u} = \mathbf{i} + \mathbf{j} \) by 3: \( 3\mathbf{u} = 3\mathbf{i} + 3\mathbf{j} \). Then, find \( 4\mathbf{v} \) by multiplying \( \mathbf{v} = \mathbf{i} - \mathbf{j} \) by 4: \( 4\mathbf{v} = 4\mathbf{i} - 4\mathbf{j} \). Finally, subtract these: \( 3\mathbf{u} - 4\mathbf{v} = (3\mathbf{i} + 3\mathbf{j}) - (4\mathbf{i} - 4\mathbf{j}) = (3 - 4)\mathbf{i} + (3 + 4)\mathbf{j} = -\mathbf{i} + 7\mathbf{j} \).

Key Concepts

Scalar Multiplication

Scalar multiplication is an essential vector operation that scales a vector by a real number, often referred to as a 'scalar.'
This operation involves multiplying each component of the vector by the scalar.
For example, if we have a vector \( \mathbf{a} = a_1 \mathbf{i} + a_2 \mathbf{j} \) and we scale it by a scalar \( c \), the result is \( c\mathbf{a} = c a_1 \mathbf{i} + c a_2 \mathbf{j} \).
This concept is useful when we need to stretch or shrink vectors, or when dealing with directional quantities in physics and engineering.

In our exercise, we apply scalar multiplication to find both \( 2 \mathbf{u} \) and \( -3 \mathbf{v} \).

• \( 2 \mathbf{u} \) is obtained by multiplying each component of \( \mathbf{u} = \mathbf{i} + \mathbf{j} \) by 2, resulting in \( 2\mathbf{u} = 2 \mathbf{i} + 2 \mathbf{j} \).
• Similarly, \( -3 \mathbf{v} \) results from multiplying each component of \( \mathbf{v} = \mathbf{i} - \mathbf{j} \) by -3, leading to \(-3\mathbf{v} = -3 \mathbf{i} + 3 \mathbf{j} \).

In both cases, the vectors maintain their direction, but their magnitude is altered, either expanded or contracted, depending on the scalar's absolute value.

Vector Addition

Vector addition is a fundamental operation that combines two or more vectors to create a resultant vector.
This is achieved by adding the corresponding components of the vectors.
Given vectors \( \mathbf{a} = a_1 \mathbf{i} + a_2 \mathbf{j} \) and \( \mathbf{b} = b_1 \mathbf{i} + b_2 \mathbf{j} \), the sum is \( \mathbf{a} + \mathbf{b} = (a_1 + b_1)\mathbf{i} + (a_2 + b_2)\mathbf{j} \).
This operation is vital in many geometry and physics applications where we need to find a combined effect of multiple vectors.

In our given problem, we calculate \( \mathbf{u} + \mathbf{v} \) by combining each vector's components:

• Start with \( \mathbf{u} = \mathbf{i} + \mathbf{j} \).
• Then, add \( \mathbf{v} = \mathbf{i} - \mathbf{j} \).
• The resultant vector \( \mathbf{u} + \mathbf{v} = (\mathbf{i} + \mathbf{j}) + (\mathbf{i} - \mathbf{j}) = 2\mathbf{i} + 0\mathbf{j} = 2\mathbf{i} \).

This example illustrates how adding vectors can lead to simplifications, such as the disappearance of the \( \mathbf{j} \) component due to the cancellation effect.

Linear Combinations

Linear combinations are a concept in linear algebra where vectors are combined using scalar multiplications and additions.
This allows us to express complex operations between vectors through simpler expressions.
A linear combination of vectors \( \mathbf{a} = a_1 \mathbf{i} + a_2 \mathbf{j} \) and \( \mathbf{b} = b_1 \mathbf{i} + b_2 \mathbf{j} \) with scalars \( c \) and \( d \) is represented as \( c\mathbf{a} + d\mathbf{b} = (ca_1 + db_1)\mathbf{i} + (ca_2 + db_2)\mathbf{j} \).

In our exercise, we tackle the combination \( 3\mathbf{u} - 4\mathbf{v} \):

• Firstly, calculate \( 3\mathbf{u} \) by scaling \( \mathbf{u} = \mathbf{i} + \mathbf{j} \) to get \( 3\mathbf{i} + 3\mathbf{j} \).
• Then, compute \( 4\mathbf{v} \) from \( \mathbf{v} = \mathbf{i} - \mathbf{j} \) to obtain \( 4\mathbf{i} - 4\mathbf{j} \).
• Finally, blend these results: \( 3\mathbf{u} - 4\mathbf{v} = (3\mathbf{i} + 3\mathbf{j}) - (4\mathbf{i} - 4\mathbf{j}) = -\mathbf{i} + 7\mathbf{j} \).

This showcases how linear combinations help in deriving new vectors with desired attributes in both mathematics and its applications in data sciences and physics.