\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:

📰 Learn the Secret Irish Phrase That Sounds Exactly Like an English Word 📰 How One Hidden Irish Influence Transformed Modern English Forever 📰 You Won’t Believe What Irish Language Secrets Hide Inside English Words 📰 Country Club In Greenville 1614499 📰 Wait Perhaps The Question Allows Non Integer 8217326 📰 Banks Around Me 6791580 📰 Figma Ticker 4931316 📰 Breathtaking Word Pictures That Pull You Into A Whole New Emotional World 1150559 📰 Emily Nutleys Secret Sohc That Will Change Everything You Know About Her 5558235 📰 What Is A Chos Wave 8808802 📰 Can You Access Alabama Power Solve Your Login Issues Quickly 8993582 📰 Sexy Movies That Will Make You Sweat The Hottest Picks You Need To Watch Now 1571711 📰 Live Deep Dive Madrids Stars Vs Legans Tacticslineups You Wont Believe 7761922 📰 Define Recursive 6065893 📰 Get Your Java Development Kit Downloaded Todayfree Jumpstart Your Projects Now 968236 📰 You Wont Believe How Dude Clickers Blast Your Gaming Speed In A Click Click To See 7418047 📰 K55 Rgb Software 386162 📰 Emerson Stock Shocking Surge Investors Are Rushing To Buy Before It Blows Away 9372934

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.