Problem 17 - 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 \(\mathbf{v} .\) $$ \mathbf{u}=\langle 2,7\rangle, \quad \mathbf{v}=\langle 3,1\rangle $$

Step-by-Step Solution

Answer

\( 2\mathbf{u} = \langle 4, 14 \rangle, -3\mathbf{v} = \langle -9, -3 \rangle, \mathbf{u}+\mathbf{v} = \langle 5, 8 \rangle, 3\mathbf{u}-4\mathbf{v} = \langle -6, 17 \rangle \).

Step 1: Calculating Scalar Multiplication of \(\mathbf{u}\)

To find \(2 \mathbf{u}\), multiply each component of \(\mathbf{u}\) by 2. Given \(\mathbf{u} = \langle 2, 7 \rangle\), we get: \(2 \mathbf{u} = 2 \times \langle 2, 7 \rangle = \langle 2 \times 2, 2 \times 7 \rangle = \langle 4, 14 \rangle\).

Step 2: Calculating Scalar Multiplication of \(-3 \mathbf{v}\)

To find \(-3 \mathbf{v}\), multiply each component of \(\mathbf{v}\) by -3. Given \(\mathbf{v} = \langle 3, 1 \rangle\), we have: \(-3 \mathbf{v} = -3 \times \langle 3, 1 \rangle = \langle -3 \times 3, -3 \times 1 \rangle = \langle -9, -3 \rangle\).

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

To find \(\mathbf{u} + \mathbf{v}\), add the corresponding components of the vectors. Given \(\mathbf{u} = \langle 2, 7 \rangle\) and \(\mathbf{v} = \langle 3, 1 \rangle\), we get: \(\mathbf{u} + \mathbf{v} = \langle 2 + 3, 7 + 1 \rangle = \langle 5, 8 \rangle\).

Step 4: Calculating Expression \(3 \mathbf{u} - 4 \mathbf{v}\)

First, find \(3 \mathbf{u}\): \(3 \times \langle 2, 7 \rangle = \langle 6, 21 \rangle\). Then, find \(4 \mathbf{v}\): \(4 \times \langle 3, 1 \rangle = \langle 12, 4 \rangle\). Subtract \(4 \mathbf{v}\) from \(3 \mathbf{u}\): \(3 \mathbf{u} - 4 \mathbf{v} = \langle 6, 21 \rangle - \langle 12, 4 \rangle = \langle 6 - 12, 21 - 4 \rangle = \langle -6, 17 \rangle\).

Key Concepts

Scalar Multiplication

Scalar multiplication is a fundamental operation in vector arithmetic. It involves multiplying each component of a vector by a scalar (a real number). This operation changes the magnitude of the vector without altering its direction, unless the scalar is negative, which reverses the direction.

For example, consider the vector \( \mathbf{u} = \langle 2, 7 \rangle \). If we want to find \( 2 \mathbf{u} \), we multiply each component by 2: \( 2 \mathbf{u} = \langle 2 \times 2, 2 \times 7 \rangle = \langle 4, 14 \rangle \). If we use a negative scalar, like \(-3\) with vector \( \mathbf{v} = \langle 3, 1 \rangle \), the resulting vector is \(-3 \mathbf{v} = \langle -3 \times 3, -3 \times 1 \rangle = \langle -9, -3 \rangle \).

Scalar multiplication is useful for scaling vectors, which can represent quantities like velocity or force in a scaled-up or scaled-down manner based on the scenario.

Vector Addition

Vector addition is another essential operation that combines two vectors into a new vector. The key here is to add corresponding components of the vectors to form a resultant vector.

For vectors \( \mathbf{u} = \langle 2, 7 \rangle \) and \( \mathbf{v} = \langle 3, 1 \rangle \), their sum is \( \mathbf{u} + \mathbf{v} = \langle 2 + 3, 7 + 1 \rangle = \langle 5, 8 \rangle \).

This operation is visually represented by joining the vectors tail-to-tip. The resultant vector, or the sum, stretches from the tail of the first vector to the tip of the second vector. Vector addition is crucial in physics and engineering, where forces or velocities are combined.

Linear Combination

A linear combination involves adding two or more scaled vectors. It's like mixing "recipes" of different vectors to get a new vector.

To compute a linear combination, we first scale the vectors by given coefficients and then add them. For instance, the expression \( 3 \mathbf{u} - 4 \mathbf{v} \) combines vectors \( 3 \times \langle 2, 7 \rangle \) and \(-4 \times \langle 3, 1 \rangle \). Calculating these gives \( 3 \mathbf{u} = \langle 6, 21 \rangle \) and \( -4 \mathbf{v} = \langle -12, -4 \rangle \). Adding these results in \( 3 \mathbf{u} - 4 \mathbf{v} = \langle 6 - 12, 21 - 4 \rangle = \langle -6, 17 \rangle \).

Linear combinations are powerful in various fields, ranging from computer graphics to solving systems of linear equations, making them an indispensable tool in mathematics.

Vectors in the Plane

Vectors in the plane are vectors that have two components, often represented as an "arrow" pointing from one point to another in a 2-dimensional coordinate system.

These vectors are typically expressed using angle brackets, such as \( \langle x, y \rangle \), where \(x\) and \(y\) are components along the x-axis and y-axis, respectively. For example, for vector \( \mathbf{u} = \langle 2, 7 \rangle \), the 2 is the component along the x-axis, while 7 is along the y-axis.

Vectors can be used to describe numerous physical phenomena, like displacement, velocity, and force. They are crucial in areas such as physics, engineering, and computer graphics for modeling movements and forces within a plane. Understanding vectors in the plane is foundational for subjects that rely heavily on spatial reasoning and geometry.