Question

If \(\mathrm{A}^{\rightarrow}=2 \mathrm{i} \wedge+5 \mathrm{j} \wedge-\mathrm{k} \wedge\) and \(\mathrm{B}^{\rightarrow}=3 \mathrm{i} \wedge-2 \mathrm{j} \wedge-4 \mathrm{k} \wedge\) the angle between \(\mathrm{A}^{\rightarrow}\) and \(\mathrm{B}^{\rightarrow}\) is (A) 0 (B) \((\pi / 2)\) (C) \((\pi / 4)\) (D) \((\pi / 6)\)

Short answer

(B) \(\frac{\pi}{2}\)

Step-by-step solution

Find the dot product of A and B

To find the angle between A and B, we first need to find the dot product of the two vectors. The dot product of A and B can be found by multiplying corresponding components of the two vectors and adding the results together: \(A \cdot B = (2i + 5j - k) \cdot (3i - 2j - 4k) \)

Calculate the dot product

Now we can multiply the components and sum them up: \(A \cdot B = (2 \times 3) + (5 \times -2) + (-1 \times -4) = 6 - 10 + 4 = 0\)

Calculate the magnitude of A and B

Next, we need to find the magnitudes of the vectors A and B. The magnitude of a vector is given by the square root of the sum of the squares of its components: \(|A| = \sqrt{(2)^{2} +(5)^{2} +(-1)^{2} } = \sqrt{30} \) \(|B| = \sqrt{(3)^{2} +(-2)^{2} +(-4)^{2} } = \sqrt{29} \)

Use the dot product formula to find the angle

Now we can use the dot product formula to find the angle θ between the two vectors. The dot product formula is: \(A \cdot B = |A| |B| \cos(\theta) \) Since we already found that \(A \cdot B = 0\), the formula becomes: \(0 = \sqrt{30} \sqrt{29} \cos(\theta)\)

Solve for θ

To find θ, we can divide both sides of the formula by \(\sqrt{30} \sqrt{29}\): \(\cos(\theta) = \frac{0}{\sqrt{30} \sqrt{29}} = 0 \) Angle θ is the angle at which cosine is 0. We know that \(\cos(\frac{\pi}{2}) = 0\), so the angle between A and B is: \(\theta = \frac{\pi}{2}\) Answer: (B) \(\frac{\pi}{2}\)

Key concepts

Vector Magnitude

The magnitude of a vector is essentially a measure of how long it is. For a 3D vector like \( \mathbf{A} = (2i, 5j, -k) \), the magnitude is found by squaring each component, adding those squared values, and then taking the square root of that total.
Here's how:

• Square each component: \(2^2 = 4\), \(5^2 = 25\), \((-1)^2 = 1\)
• Add the squared values: \(4 + 25 + 1 = 30\)
• Take the square root of the sum: \(|A| = \sqrt{30}\)

This gives us the vector's magnitude, which is its length or the distance from the origin to its endpoint in a vector coordinate system.
The concept of magnitude is fundamental in linear algebra as it helps in determining the vector's size and comparing it with other vectors. Similarly, for vector \( \mathbf{B} = (3i, -2j, -4k) \), the magnitude is calculated as \( \sqrt{29} \) following the same steps.

Cosine of Angle Between Vectors

The cosine of the angle between two vectors is crucial in determining how those vectors relate to each other spatially. To find this cosine, the dot product formula is used, which relates the dot product of the vectors to their magnitudes and the cosine of the angle between them.
The formula is: \( A \cdot B = |A| |B| \cos(\theta) \).

• The dot product \(A \cdot B\) gives us a scalar value which, in this case, was found to be \(0\).
• The magnitudes \(|A|\) and \(|B|\) were calculated as \(\sqrt{30}\) and \(\sqrt{29}\), respectively.

Since the dot product is zero, the formula simplifies to \(\cos(\theta) = 0\). This indicates that the angle \(\theta\) between the vectors is specifically \(\frac{\pi}{2}\) radians or 90 degrees, meaning they are perpendicular.

Angle Between Vectors

Finding the angle between vectors is a key task in linear algebra, often achieved through the dot product. When determining this angle, the goal is to understand how the vectors interact or relate spatially.
For vectors \( \mathbf{A} = 2i + 5j - k \) and \( \mathbf{B} = 3i - 2j - 4k \), after finding their dot product and magnitudes, we observe

• If the dot product is zero, the vectors are perpendicular, as cosine \(\theta = 0\)
• If vectors are parallel, cosine \(\theta\) would be \(\pm 1\)

In our case, with \(\theta\) being \(\frac{\pi}{2}\), this signifies they meet at a right angle. Understanding angles between vectors helps in determining the geometric properties of vector spaces.

Linear Algebra Concepts

Linear algebra is a branch of mathematics focusing on vector spaces and linear mappings between them, playing a significant role in understanding spatial relationships and transformations. Core concepts include vectors, vector operations like the dot product, and magnitude measurement.

• Vectors are fundamental building blocks used to express direction and magnitude.
• The dot product assesses how much of one vector goes in the direction of another.
• Magnitude offers a measure of the length or size of a vector.

Understanding these concepts aids in solving complex problems, be it in physics with forces or computer graphics with object positioning. The study of linear algebra allows one to model and analyze multi-dimensional spaces, essential in fields like machine learning, geometry, and computer science. Utilizing these principles helps clarify the properties and interactions of different vector elements within these spaces.