Problem 31 - Algebra and Trigonometry

Question

\(31-36\) 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: Find $2 \mathbf{u}$

To find \(2 \mathbf{u}\), we multiply each component of the vector \(\mathbf{u}\) by 2. Since \(\mathbf{u} = \langle 2, 7 \rangle\), 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: Find $-3 \mathbf{v}$

To find \(-3 \mathbf{v}\), we multiply each component of the vector \(\mathbf{v}\) by -3. Since \(\mathbf{v} = \langle 3, 1 \rangle\), we get: \[-3 \mathbf{v} = -3 \times \langle 3, 1 \rangle = \langle -3 \times 3, -3 \times 1 \rangle = \langle -9, -3 \rangle.\]

Step 3: Find $\mathbf{u} + \mathbf{v}$

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

Step 4: Find $3 \mathbf{u} - 4 \mathbf{v}$

First, calculate \(3 \mathbf{u}\) by multiplying each component of \(\mathbf{u}\) by 3:\[3 \mathbf{u} = 3 \times \langle 2, 7 \rangle = \langle 6, 21 \rangle.\]Next, calculate \(4 \mathbf{v}\) by multiplying each component of \(\mathbf{v}\) by 4:\[4 \mathbf{v} = 4 \times \langle 3, 1 \rangle = \langle 12, 4 \rangle.\]Finally, find \(3 \mathbf{u} - 4 \mathbf{v}\) by subtracting the components of \(4 \mathbf{v}\) from those of \(3 \mathbf{u}\):\[3 \mathbf{u} - 4 \mathbf{v} = \langle 6 - 12, 21 - 4 \rangle = \langle -6, 17 \rangle.\]

Key Concepts

Scalar Multiplication

Scalar multiplication involves scaling a vector by a number, known as a scalar. In scalar multiplication, each component of the vector is multiplied by this scalar. For example, if we have a vector \( \mathbf{u} = \langle 2, 7 \rangle \) and a scalar 2, then multiplying the vector by this scalar results in the vector \( 2 \mathbf{u} = \langle 4, 14 \rangle \). This operation stretches the vector if the scalar is greater than 1, shrinks it if it's between 0 and 1, or reverses its direction if the scalar is negative. Scalar multiplication is crucial because it allows you to adjust the magnitude of a vector without changing its direction, unless the scalar is negative, in which case the direction is reversed.

Vector Addition

Vector addition is the process of combining two vectors to form a third vector. This new vector is called the resultant vector. To add two vectors, you simply add their corresponding components. For vectors \( \mathbf{u} = \langle 2, 7 \rangle \) and \( \mathbf{v} = \langle 3, 1 \rangle \), the sum is calculated as \( \mathbf{u} + \mathbf{v} = \langle 2 + 3, 7 + 1 \rangle = \langle 5, 8 \rangle \). This means that the resultant vector takes a path that moves from the origin to point \( \langle 5, 8 \rangle \), which effectively combines the directions and magnitudes of both original vectors. Vector addition is key in various applications like physics, for determining forces acting on an object, or navigation, where combining paths is necessary.

Vector Subtraction

Vector subtraction is similar to vector addition, but instead of adding the corresponding components, you subtract them. This operation is useful for finding the difference between two vectors or determining the vector you would need to add to reach from the tip of one vector to the tip of another. When subtracting vector \( \mathbf{v} = \langle 3, 1 \rangle \) from \( 3 \mathbf{u} = \langle 6, 21 \rangle \), you compute it as \( 3 \mathbf{u} - 4 \mathbf{v} = \langle 6 - 12, 21 - 4 \rangle = \langle -6, 17 \rangle \). This provides a new vector that positions you at a point relative to where you started.

Vectors

Vectors are mathematical objects characterized by both magnitude and direction. Unlike numbers, which only have magnitude, vectors describe quantities like velocity, force, and displacement, which have more than just size—they also have a direction. Vectors are usually depicted as arrows with the length representing the magnitude and the arrowhead indicating the direction. For instance, if \( \mathbf{u} = \langle 2, 7 \rangle \) and \( \mathbf{v} = \langle 3, 1 \rangle \), these vectors can also be visualized geometrically as arrows starting from the origin on a Cartesian plane. Understanding vectors is fundamental in fields such as physics for modeling forces, engineering for structural analysis, and computer graphics for manipulating images.