\langle 1, 0, 4 \rangle \times \langle 2, -1, 3 \rangle = \langle (0)(3) - (4)(-1), -[(1)(3) - (4)(2)], (1)(-1) - (0)(2) \rangle = \langle 0 + 4, -(3 - 8), -1 - 0 \rangle = \langle 4, 5, -1 \rangle

Understanding Cross Products in 3D Space: A Step-by-Step Calculation of ⟨1, 0, 4⟩ × ⟨2, −1, 3⟩

The cross product of two vectors in three-dimensional space is a fundamental operation in linear algebra, physics, and engineering. Despite its seemingly abstract appearance, the cross product produces another vector perpendicular to the original two. This article explains how to compute the cross product of the vectors ⟨1, 0, 4⟩ and ⟨2, −1, 3⟩ using both algorithmic step-by-step methods and component-wise formulas—ultimately revealing why ⟨1, 0, 4⟩ × ⟨2, −1, 3⟩ = ⟨4, 5, −1⟩.

Understanding the Context

What Is a Cross Product?

Given two vectors a = ⟨a₁, a₂, a₃⟩ and b = ⟨b₁, b₂, b₃⟩ in ℝ³, their cross product a × b is defined as:

⟨a₂b₃ − a₃b₂,
−(a₁b₃ − a₃b₁),
a₁b₂ − a₂b₁⟩

This vector is always orthogonal to both a and b, and its magnitude equals the area of the parallelogram formed by a and b.

$\langle 1, 0, 4 \rangle \times \langle 2, -1, 3 \rangle = \langle (0)(3) - (4)(-1), -[(1)(3) - (4)(2)], (1)(-1) - (0)(2) \rangle = \langle 0 + 4, -(3 - 8), -1 - 0 \rangle = \langle 4, 5, -1 \rangle$

Key Insights

Applying the Formula to ⟨1, 0, 4⟩ × ⟨2, −1, 3⟩

Let a = ⟨1, 0, 4⟩ and b = ⟨2, −1, 3⟩.

Using the standard cross product formula:

Step 1: Compute the first component
(0)(3) − (4)(−1) = 0 + 4 = 4

🔗 Related Articles You Might Like:

📰 This Game Brings Michael Jackson’s The Experience to Life—Could You Survive It?" 📰 You Won’t Believe How This MW3 Update Changed the Game Forever! 📰 MW3 Mastery Secrets You Need to See—Game Perms Exploded in Popularity! 📰 Betsey Johnson Wedding Shoes 📰 Betsy Braddock 📰 Betta In Fish Tank 📰 Betta Tank Mates 📰 Better Build Buy Sims 4 📰 Better Call A Saul 📰 Better Call Saul Season 3 📰 Better Call Saul Season 5 📰 Better Call Saul Series 3 📰 Better Call Saul Tv Show Season 3 📰 Better Exceptions Sims 4 📰 Better Saul Season 3 📰 Better Than Paper 📰 Better To Have Loved And Lost 📰 Better Wild

Final Thoughts

Step 2: Compute the second component
−[(1)(3) − (4)(2)] = −[3 − 8] = −[−5] = 5

Step 3: Compute the third component
(1)(−1) − (0)(2) = −1 − 0 = −1

Putting it all together:
⟨1, 0, 4⟩ × ⟨2, −1, 3⟩ = ⟨4, 5, −1⟩

Why Does This Work? Intuition Behind the Cross Product

The cross product’s components follow the determinant of a 3×3 matrix with unit vectors and the vector components:

⟨i, j, k⟩
| 1 0 4
|² −1 3

Expanding the determinant:

i-component: (0)(3) − (4)(−1) = 0 + 4 = 4
j-component: −[(1)(3) − (4)(2)] = −[3 − 8] = 5
k-component: (1)(−1) − (0)(2) = −1 − 0 = −1

This confirms that the formula used is equivalent to the cofactor expansion method, validating the result.