Problem 1 - Calculus for Biology and Medicine

Question

Let \(\mathbf{x}=[1,0,-1]^{\prime}\) and \(\mathbf{y}=[-2,1,0]^{\prime}\). (a) Find \(\mathbf{x}+\mathbf{y}\). (b) Find \(2 \mathbf{x}\). (c) Find \(-3 \mathbf{y}\).

Step-by-Step Solution

Answer

(a) \([-1, 1, -1]\), (b) \([2, 0, -2]\), (c) \([6, -3, 0]\).

Step 1: Understanding Vector Addition (Finding \(\mathbf{x}+\mathbf{y}\))

To find the sum of two vectors \(\mathbf{x}\) and \(\mathbf{y}\), we add corresponding components of the two vectors. In our case, we compute \([1 + (-2), 0 + 1, -1 + 0]\) which gives the resulting vector.

Step 2: Compute \(\mathbf{x}+\mathbf{y}\)

Perform the component-wise addition: \[\mathbf{x}+\mathbf{y} = [1 + (-2), 0 + 1, -1 + 0] = [-1, 1, -1]\] Thus, the result is \([-1, 1, -1]\).

Step 3: Understanding Scalar Multiplication (Finding \(2 \mathbf{x}\))

To multiply a vector by a scalar, we multiply each component of the vector by the scalar. Here, we will multiply each component of \(\mathbf{x}\) by 2.

Step 4: Compute \(2 \mathbf{x}\)

Perform the scalar multiplication:\[2 \mathbf{x} = 2 \times [1, 0, -1] = [2 \times 1, 2 \times 0, 2 \times (-1)] = [2, 0, -2]\]Thus, \(2 \mathbf{x} = [2, 0, -2]\).

Step 5: Understanding Scalar Multiplication (Finding \(-3 \mathbf{y}\))

Similarly, to find \(-3 \mathbf{y}\), multiply each component of \(\mathbf{y}\) by \(-3\).

Step 6: Compute \(-3 \mathbf{y}\)

Perform the scalar multiplication:\[-3 \mathbf{y} = -3 \times [-2, 1, 0] = [-3 \times (-2), -3 \times 1, -3 \times 0] = [6, -3, 0]\]Thus, \(-3 \mathbf{y} = [6, -3, 0]\).

Key Concepts

Vector Addition

Vector addition is a basic yet pivotal operation in vector algebra. It involves adding two vectors to produce a new vector, often called the resultant vector. To add vectors, you simply add their corresponding components.

For example, consider two vectors \( \mathbf{x} = [1, 0, -1] \) and \( \mathbf{y} = [-2, 1, 0] \). To find the vector sum \( \mathbf{x} + \mathbf{y} \), perform the following operations on each component:

• First component: \( 1 + (-2) = -1 \)
• Second component: \( 0 + 1 = 1 \)
• Third component: \( -1 + 0 = -1 \)

After calculating each component individually, combine them to form the resultant vector: \( [-1, 1, -1] \). This process provides a straightforward approach to determine the sum of vectors in any dimension.

Scalar Multiplication

Scalar multiplication is a fundamental vector operation. It involves multiplying each component of a vector by a scalar (a constant value, not a vector). This operation scales the vector, affecting its length but not its direction if the scalar is positive.

For illustration, if you have the vector \( \mathbf{x} = [1, 0, -1] \), and you want to find \( 2 \mathbf{x} \), multiply each component by 2:

• Multiply the first component: \( 2 \times 1 = 2 \)
• Multiply the second component: \( 2 \times 0 = 0 \)
• Multiply the third component: \( 2 \times (-1) = -2 \)

The resulting vector after this operation is \( [2, 0, -2] \).

Similarly, for the vector \( \mathbf{y} = [-2, 1, 0] \), to find \(-3 \mathbf{y} \), multiply each component by \(-3\):

• First component: \(-3 \times (-2) = 6 \)
• Second component: \(-3 \times 1 = -3 \)
• Third component: \(-3 \times 0 = 0 \)

This gives the vector \( [6, -3, 0] \). Notice how a negative scalar not only changes the magnitude but also reverses the direction of the vector.

Vector Components

Understanding vector components is key to performing operations like addition and scalar multiplication. A vector in a three-dimensional space is represented as \([x, y, z]\), where \(x\), \(y\), and \(z\) represent its components along the respective axes.

Consider a vector \( \mathbf{v} = [x, y, z] \). Each part of this vector is known as a component:

• \(x\) is the component along the x-axis.
• \(y\) is the component along the y-axis.
• \(z\) is the component along the z-axis.

The components determine the direction and magnitude of the vector.

When performing vector operations, maintaining accuracy with components is essential. Each component is treated separately in addition or scalar multiplication. For instance, when adding two vectors or when a vector is scaled by a scalar, each component undergoes the operation individually, ensuring that the vector's overall characteristics are correctly computed.