Question

Find a vector perpendicular to both \(\mathbf{i}-3 \mathbf{j}+2 \mathbf{k}\) and \(5 \mathbf{i}-\mathbf{j}-4 \mathbf{k}\).

Short answer

14 ( \textbf{i} + \textbf{j} + \textbf{k} )

Step-by-step solution

Understand the Problem

Finding a vector that is perpendicular to two given vectors involves finding a vector that is orthogonal to both. This can be achieved using the cross product of the two vectors.

Write Down the Given Vectors

Let the first vector be \(\textbf{A} = \textbf{i} - 3 \textbf{j} + 2 \textbf{k} \) and the second vector be \(\textbf{B} = 5 \textbf{i} - \textbf{j} - 4 \textbf{k} \).

Set Up the Cross Product Formula

The cross product of \(\textbf{A}\) and \(\textbf{B}\) is given by \(\textbf{A} \times \textbf{B} = (a_2b_3 - a_3b_2) \textbf{i} + (a_3b_1 - a_1b_3) \textbf{j} + (a_1b_2 - a_2b_1) \textbf{k}\).

Substitute the Vector Components

Substituting \(a_1 = 1, a_2 = -3, a_3 = 2 \), and \(b_1 = 5, b_2 = -1, b_3 = -4\) into the cross product formula: \(\textbf{A} \times \textbf{B} = ((-3)(-4) - (2)(-1)) \textbf{i} + ((2)(5) - (1)(-4)) \textbf{j} + ((1)(-1) - (-3)(5)) \textbf{k} \).

Simplify the Calculation

Calculate each component: \( \textbf{A} \times \textbf{B} = (12 + 2) \textbf{i} + (10 + 4) \textbf{j} + (-1 + 15) \textbf{k} \). This gives \( \textbf{A} \times \textbf{B} = 14 \textbf{i} + 14 \textbf{j} + 14 \textbf{k} \).

Write the Final Answer

The vector perpendicular to both given vectors can be written as \( \textbf{A} \times \textbf{B} = 14 (\textbf{i} + \textbf{j} + \textbf{k}) \).

Key concepts

Orthogonal Vectors

Orthogonal vectors are vectors that meet at a right angle (90 degrees). When two vectors are orthogonal, their dot product is zero. Knowing this can help you understand why the cross product results in a vector perpendicular to both original vectors. To check orthogonality, find the dot product of the vectors. If you get zero, the vectors are orthogonal. For example, let \(\textbf{u} = \textbf{i} \) and \(\textbf{v} = \textbf{j} \). Their dot product is 0, confirming they are orthogonal.

Perpendicular Vector

A perpendicular vector is a vector that forms a 90-degree angle with another vector. In 3D space, this concept extends to finding a vector that is perpendicular to two given vectors. This is done using the cross product operation. For example, in the given exercise, the cross product of \( \textbf{i} - 3 \textbf{j} + 2 \textbf{k} \) and \( 5 \textbf{i} - \textbf{j} - 4 \textbf{k} \) results in a vector \( 14 (\textbf{i} + \textbf{j} + \textbf{k}) \). This new vector is perpendicular (or orthogonal) to both original vectors.

Vector Calculations

Vector calculations involve various operations like addition, subtraction, dot product, and cross product.
The cross product specifically helps us find a vector that is orthogonal to two given vectors. This is especially useful in physics and engineering. To perform a cross product, use the determinant of a matrix composed of the unit vectors and the components of the original vectors. In the exercise, we substitute the components of \(\textbf{A} = \textbf{i} - 3 \textbf{j} + 2 \textbf{k} \) and \(\textbf{B} = 5 \textbf{i} - \textbf{j} - 4 \textbf{k} \) into the cross product formula and simplify to find the perpendicular vector.

Vector Operation

Vector operations are fundamental in vector math. One important operation is the cross product, used to find a vector perpendicular to two others.
The cross product formula is: \( \textbf{A} \times \textbf{B} = (a_2b_3 - a_3b_2) \textbf{i} + (a_3b_1 - a_1b_3) \textbf{j} + (a_1b_2 - a_2b_1) \textbf{k} \).
As seen in the exercise, this operation involves substituting the vector components into the formula and simplifying. The final vector \( \textbf{A} \times \textbf{B} = 14 (\textbf{i} + \textbf{j} + \textbf{k}) \) represents a new vector orthogonal to both \( \textbf{A} \) and \( \textbf{B} \). This is a powerful tool for many applications, including physics and engineering problems.