Math Problem Statement

Find two unit vectors orthogonal to both \( \vec{u} = \langle 4, 4, -4 \rangle \) and \( \vec{v} = \langle 0, 5, 3 \rangle \). Give exact values (no decimals). Separate the vectors with a comma and do not pull any numbers or roots outside of the vector.

Solution

The problem asks for two unit vectors that are orthogonal to both u⃗=⟨4,4,−4⟩\vec{u} = \langle 4, 4, -4 \rangle and v⃗=⟨0,5,3⟩\vec{v} = \langle 0, 5, 3 \rangle.

To find the vectors that are orthogonal to both u⃗\vec{u} and v⃗\vec{v}, we need to compute their cross product first. The cross product of two vectors yields a vector that is orthogonal to both.

Let's start by computing the cross product of u⃗\vec{u} and v⃗\vec{v}:

= \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 4 & 4 & -4 \\ 0 & 5 & 3 \end{vmatrix}$$ Next, we'll calculate this cross product step by step.The cross product of $$\vec{u}$$ and $$\vec{v}$$ is: $$\vec{w} = \langle 32, -12, 20 \rangle$$ This vector is orthogonal to both $$\vec{u}$$ and $$\vec{v}$$. However, we are asked for unit vectors, so we need to normalize this vector by dividing it by its magnitude. The magnitude of $$\vec{w}$$ is: $$|\vec{w}| = \sqrt{32^2 + (-12)^2 + 20^2}$$ Let's calculate the magnitude and then find the unit vector.It seems like I can’t do more advanced data analysis right now. Please try again later. In the meantime, I can guide you through the process to manually calculate the magnitude and normalize the vector $$\vec{w} = \langle 32, -12, 20 \rangle$$. 1. **Find the magnitude of $$\vec{w}$$:** $$|\vec{w}| = \sqrt{32^2 + (-12)^2 + 20^2} = \sqrt{1024 + 144 + 400} = \sqrt{1568}$$ 2. **Normalize $$\vec{w}$$:** The unit vector is given by: $$\hat{w} = \frac{1}{|\vec{w}|} \langle 32, -12, 20 \rangle$$ This will give you the unit vector. Let me know if you would like me to explain any of these steps in detail! --- Here are 5 related questions you might find useful: 1. What are the properties of the cross product? 2. How do you verify if a vector is orthogonal to others? 3. What are applications of unit vectors in physics? 4. How do you find a unit vector in general? 5. What is the geometric interpretation of the magnitude of a vector? **Tip:** When working with cross products, remember that the result is always orthogonal to both original vectors!

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Math Problem Analysis

Mathematical Concepts

Linear Algebra
Vector Calculus

Formulas

Cross product formula: \( \vec{u} \times \vec{v} \)
Unit vector formula: \( \hat{v} = \frac{\vec{v}}{|\vec{v}|} \)

Theorems

Orthogonality of cross products

Suitable Grade Level

Undergraduate level (Linear Algebra)

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