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

Continue Reading

\frac{x^2 + 1}{x^2 - 1} + \frac{x^2 - 1}{x^2 + 1} = \frac{(x^2 + 1)^2 + (x^2 - 1)^2}{(x^2 - 1)(x^2 + 1)} = \frac{2x^4 + 2}{x^4 - 1} = 2 \cdot \frac{x^4 + 1}{x^4 - 1}
Let’s compute $ x^4 + 1 $, $ x^4 - 1 $:
We know $ x^2 + \frac{1}{x^2} = 23 \Rightarrow x^4 + \frac{1}{x^4} = 527 $, so:

🔗 Related Articles You Might Like:

📰 Shocking Realities Emerging at Red Ribbon Week 2025 That Everyone Should See Immediately 📰 You Won’t Believe What One Mechanic Uses for Peak Engine Performance 📰 This Hidden Truth About Reliable Automotive Parts Will Shock You 📰 Secrets Of The Selkie Dress That Will Blow Your Mind 📰 Secrets Of The September Birth Flower Why Its Terrifyingly Special For You 📰 Secrets Rebecca J Revealed In Private Porn Nightmares No One Talks About 📰 Secrets Revealed Draw Desktop J Perfect Seashell Like A Pro 📰 Secrets Revealed Raquel Welchs Stunning Unveiling In Exclusive Nude Revelation 📰 Secrets Revealed What Saginaw County Jails Files Cant Hide 📰 Secrets Rising From The Fjords Scotlands Bodies Of Water That Hold The Truth 📰 Secrets Saltwater Enthusiasts Are Desperate To Discoverdont Miss It 📰 Secrets Sammy Keeps What Her Thighs Say In The Stripping Light Shocking Truth Inside 📰 Secrets Selena Never Casually Mentions Her Weight Loss Full Breakdown 📰 Secrets Slip From Selena Vargas Facenow The Truth Is Unstoppable 📰 Secrets Spilled By Radio Pasfarda Lost Voices In The Static 📰 Secrets To Healing Faulty Repairlink Systems You Never Expected 📰 Secrets Unleashed The Forgotten Show That Powers Santa Cruz Cinemas Magic Evening 📰 Secrets Unlocked Sanctuary Dispensary Transforms Lives Forever

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.