Math Problem Statement

Find a unit vector that is orthogonal to both \( \vec{u} = (6, 2, 8) \) and \( \vec{v} = (4, -3, -1) \).

Solution

To find a unit vector that is orthogonal to both vectors u⃗=(6,2,8)\vec{u} = (6, 2, 8) and v⃗=(4,−3,−1)\vec{v} = (4, -3, -1), we can follow these steps:

Step 1: Find the Cross Product of u⃗\vec{u} and v⃗\vec{v}

The cross product u⃗×v⃗\vec{u} \times \vec{v} gives a vector that is orthogonal to both u⃗\vec{u} and v⃗\vec{v}.

\begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 6 & 2 & 8 \\ 4 & -3 & -1 \\ \end{vmatrix}$$ This determinant expands to: $$\vec{u} \times \vec{v} = \hat{i}(2 \cdot (-1) - 8 \cdot (-3)) - \hat{j}(6 \cdot (-1) - 8 \cdot 4) + \hat{k}(6 \cdot (-3) - 2 \cdot 4)$$ Calculating each component: 1. **i-component**: $$2 \cdot (-1) - 8 \cdot (-3) = -2 + 24 = 22$$ 2. **j-component**: $$-(6 \cdot (-1) - 8 \cdot 4) = -(-6 - 32) = 38$$ 3. **k-component**: $$6 \cdot (-3) - 2 \cdot 4 = -18 - 8 = -26$$ So, $$\vec{u} \times \vec{v} = (22, 38, -26)$$ ### Step 2: Normalize the Vector To make this vector a unit vector, we need to divide it by its magnitude. 1. **Magnitude of $$(22, 38, -26)$$**: $$\sqrt{22^2 + 38^2 + (-26)^2} = \sqrt{484 + 1444 + 676} = \sqrt{2604} = 51.0235$$ 2. **Unit Vector**: $$\text{Unit vector} = \frac{1}{51.0235} \times (22, 38, -26) \approx (0.43147, 0.74497, -0.50934)$$ ### Final Answer The unit vector that is orthogonal to both $$\vec{u}$$ and $$\vec{v}$$ is approximately: $$\boxed{(0.43147, 0.74497, -0.50934)}$$ Would you like further clarification, or do you have any questions on this solution? --- ### Follow-Up Questions: 1. What is the significance of the cross product in vector mathematics? 2. How do you calculate the magnitude of a vector? 3. Why is it important to normalize vectors in certain applications? 4. Can there be more than one unit vector orthogonal to two given vectors? 5. How does rounding affect the accuracy of vector calculations? ### Tip: For higher precision in calculations, avoid rounding intermediate steps too early, especially in sensitive problems.

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Math Problem Analysis

Mathematical Concepts

Vector Cross Product
Vector Magnitude
Unit Vector
Orthogonal Vectors

Formulas

Cross product of vectors \( \vec{u} \times \vec{v} \)
Magnitude of a vector \( \sqrt{x^2 + y^2 + z^2} \)
Unit vector formula \( \frac{\vec{v}}{\|\vec{v}\|} \)

Theorems

Cross Product Theorem (to find orthogonal vectors)

Suitable Grade Level

Grades 11-12 or College

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