\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:

📰 Who Watches an All White Outfit Pale Like a Ghost? This Trend is Unstoppable! 📰 Transform Your Look Instantly… The Hidden Magic of an All White Outfit! 📰 From Red Carpet to Street Style: This All White Outfit Will Shock You! 📰 Japanese Clothes Thatll Blow Your Mindwhy Every Outfit Feels Like Art 📰 Jiffy Cornbread Casserole The Secret Shortcut That Makes Weeknight Meals Irresistible 📰 Join The Action Uncover The Hottest Moments From Pokmon Gos Biggest Community Day 📰 Join The Christian R Sabat Voice Actor Phenomenon Heres What Makes Him Unstoppable 📰 Join The Cocokind Lip Balm Craze Before Its Too Latestop Scrolling 📰 Join The Costco Affiliate Program Today And Cash In Bigguaranteed Tips Inside 📰 Join The Lyoko Revolution Decoding Code Lyoko Code Like Never Before 📰 Jung Sung Bin 25513 20226712 📰 Just Discovered The Shocking Truth Behind The Chrysos Heirs Youve Never Heard Of 📰 Just Logged In Conroe Isd Sso Login Secrets Youve Been Missing 📰 Just Look Here Free Codes To Transform Your Rca Universal Remote Instantly 📰 Just Saw The Ultimate Congrats Memecongrats But Wait This Reaction Is Unusable 📰 Keep Watching These Cool Dinosaurs Are About To Blow Your Mind Yes Theyre Real 📰 Key 20 Breathtaking Couple Tattoo Ideas That Will Sweep You Off Your Feet 📰 Key To Stylish Floors The Brand Changing Common Rug Sizes You Must Know

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.