\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⟩.

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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.

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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

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⟩

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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.

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Practical Applications of Cross Products

Cross products are essential in:

• Physics: Calculating torque, angular momentum, and magnetic forces.
• Computer Graphics: Determining surface normals and lighting calculations.
• Engineering: Analyzing rotational dynamics and spatial orientation.

Understanding how to compute cross products manually enables deeper insight into 3D geometry and related applications.

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Final Recap

The cross product of ⟨1, 0, 4⟩ and ⟨2, −1, 3⟩ is correctly calculated as ⟨4, 5, −1⟩ using either:

• Step-by-step component substitution
• The determinant-based expansion method

Every coordinate is derived using standard algebraic rules, ensuring both accuracy and clarity. Whether you're studying geometry or applying vector math in programming or physics, mastering this operation is invaluable.

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Keywords: cross product, vector math, 3D vectors, cross product formula, ⟨1, 0, 4⟩ × ⟨2, −1, 3⟩, determinant method, physics vectors, computer graphics, linear algebra.

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Need to compute more cross products? Try breaking down each vector into its components and applying the formula methodically—this approach works every time and strengthens your vector calculus skills!