Problem 22 - Precalculus

Question

Perform the indicated vector operation, given \(u=(-4,3)\) and \(v=\langle 2,-5\rangle\) $$5(\mathbf{u}+\mathbf{v})$$

Step-by-Step Solution

Answer

The result is \((-10, -10)\).

Step 1: Understand Vector Notation

Recognize the vector notation. The vectors given are \( \mathbf{u} = (-4, 3) \) and \( \mathbf{v} = \langle 2, -5 \rangle \), which are equivalent notations for 2D vectors.

Step 2: Add the Vectors

Perform the addition of vectors \( \mathbf{u} \) and \( \mathbf{v} \) by adding their corresponding components. \[ \mathbf{u} + \mathbf{v} = (-4 + 2, 3 - 5) = (-2, -2) \]

Step 3: Scalar Multiplication of the Resulting Vector

Multiply the resulting vector from Step 2 by the scalar 5. Multiply each component by 5: \[ 5(-2, -2) = (5 \times -2, 5 \times -2) = (-10, -10) \]

Step 4: Conclude with the Result

The resulting vector from the operation \( 5(\mathbf{u} + \mathbf{v}) \) is \( (-10, -10) \).

Key Concepts

Scalar Multiplication

Scalar multiplication is a fundamental operation in vector mathematics. When we multiply a vector by a scalar, we scale the magnitude of the vector, but the direction remains the same. Let's break this down:

• A scalar is simply a real number that can stretch or contract a vector.
• Each component of the vector is multiplied by the scalar, altering its overall size.

For example, consider a vector \( \mathbf{a} = (x, y) \) and a scalar \( k \). The scalar multiplication is performed as:\[ k \mathbf{a} = (k \times x, k \times y) \]This operation increases or decreases the vector's length. In the given exercise, we multiplied the vector \((-2, -2)\) by 5, which resulted in \((-10, -10)\). Here, the vector's direction remained constant (both components are negative), but its magnitude increased fivefold.

Vector Addition

Vector addition involves combining two or more vectors to determine the resultant vector. This operation is very visual and intuitive if you imagine moving along each vector sequentially.

• You simply sum the corresponding components of each vector.
• This gives a new vector that effectively represents the combined force or movement of the original vectors.

For the given vectors \( \mathbf{u} = (-4, 3) \) and \( \mathbf{v} = \langle 2, -5 \rangle \), adding them looks like this:\[ \mathbf{u} + \mathbf{v} = (-4 + 2, 3 - 5) = (-2, -2) \]Notice how each component from both vectors is added separately (i.e., \(-4\) with 2, and 3 with \(-5\)). The result, \((-2, -2)\), is a new vector that combines the influences of \(\mathbf{u}\) and \(\mathbf{v}\). Understanding vector addition is crucial when dealing with forces, speeds, or any scenario represented by vectors.

2D Vectors

Two-dimensional vectors, or 2D vectors, are vectors that exist within a plane, having both an x-axis and a y-axis component. Vectors in 2D allow us to describe movement, forces, or positions in a plane in a very mathematical and visual way.

• These vectors are described using an ordered pair, such as \((x, y)\), where \(x\) is the horizontal component, and \(y\) is the vertical component.
• Vectors can be represented graphically as arrows, with the tail at the origin and the head at the point \((x, y)\).

Knowing how to work with 2D vectors is essential for solving problems in physics, engineering, and mathematics, as they help in envisioning forces and movements. Remember the vectors from our exercise, \( \mathbf{u} = (-4, 3) \) and \( \mathbf{v} = \langle 2, -5 \rangle \). Both are defined in a two-dimensional plane, facilitating straightforward calculations using basic arithmetic operations like addition and scalar multiplication. This simplification is why 2D vectors are so widely used in various practical applications.