\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:

📰 ttss movie 📰 ct what 📰 hinesville ga 📰 Never Guessed Who Spains Ready With These Players Against Netherlands Legends 7631794 📰 A Data Center Consumes 500 Kilowatt Hours Per Day Electricity Costs 012 Per Kwh If They Operate 365 Days A Year What Is The Annual Electricity Cost 9652954 📰 How Mastering Synonymal Shifts Your Life Without Ever Saying A Word 2610832 📰 Apptrace The Ultimate App Debugging Tool Thats Amazingly Simple But Powerful 2186499 📰 Epic Games Setup Ended Prematurely 6849046 📰 Waitprotein Can Reverse Fat 7 Super Power High Protein Recipes Inside 8828978 📰 Kiss Me Deadly 1955 8174266 📰 Crazy Surge Truist Stock Surpasses Expectationsis This Your Investment Opportunity 5233924 📰 The Latest Cannoo News Shock Inside The Secret Deal That Could Change Everything 7904640 📰 Download The Alexa App On Iphonevoice Commands Made Easier Download Now Transform Your Routine 9031890 📰 Mortal Kombat Vs Most Anticipated Deadly Alliance Twist Youve Been Craving 5243898 📰 You Wont Believe What This Skeleboner Can Doshocking Unity Features Youll Wish You Saw 5463572 📰 Are Xbox Servers Down 3517415 📰 Play Vita Mahjong Online Free Free Game Thats Super Addictive 965918 📰 What Is A Mega Backdoor Roth 9814279

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.