Solved Problems: Dot Product of 3D Vectors for High School Pre-Calculus

📚 Understanding the Dot Product of 3D Vectors

The dot product, also known as the scalar product, is an operation that takes two vectors and returns a scalar. For 3D vectors, it's a simple yet powerful tool with applications in physics, engineering, and computer graphics.

📜 History and Background

The concept of vectors emerged in the 19th century, with contributions from mathematicians and physicists like William Rowan Hamilton and Josiah Willard Gibbs. The dot product became an essential part of vector algebra, providing a way to quantify the alignment between two vectors.

📐 Key Principles

• ➕ Definition: For two vectors $\vec{a} = (a_1, a_2, a_3)$ and $\vec{b} = (b_1, b_2, b_3)$, their dot product is defined as: $\vec{a} \cdot \vec{b} = a_1b_1 + a_2b_2 + a_3b_3$.
• 🧮 Calculation: Multiply corresponding components and sum the results.
• ✨ Geometric Interpretation: The dot product can also be expressed as $\vec{a} \cdot \vec{b} = |\vec{a}| |\vec{b}| \cos(\theta)$, where $|\vec{a}|$ and $|\vec{b}|$ are the magnitudes of the vectors, and $\theta$ is the angle between them.
• orthogonal Orthogonality: If $\vec{a} \cdot \vec{b} = 0$, then the vectors are orthogonal (perpendicular).

⚙️ Real-world Examples

• 🏗️ Engineering: Calculating the work done by a force vector over a displacement vector. If a force $\vec{F} = (5, 2, 3)$ N acts on an object moving along a displacement $\vec{d} = (2, 1, 4)$ m, the work done is $\vec{F} \cdot \vec{d} = (5)(2) + (2)(1) + (3)(4) = 10 + 2 + 12 = 24$ Joules.
• 🎮 Computer Graphics: Determining the angle between a light source and a surface to calculate shading.
• 🛰️ Physics: Finding the component of a velocity vector in a certain direction.

📝 Practice Problems

Problem 1: Find the dot product of $\vec{a} = (1, 2, 3)$ and $\vec{b} = (4, 5, 6)$.

Solution: $\vec{a} \cdot \vec{b} = (1)(4) + (2)(5) + (3)(6) = 4 + 10 + 18 = 32$.

Problem 2: Find the dot product of $\vec{p} = (-2, 0, 1)$ and $\vec{q} = (3, -1, -5)$.

Solution: $\vec{p} \cdot \vec{q} = (-2)(3) + (0)(-1) + (1)(-5) = -6 + 0 - 5 = -11$.

Problem 3: Determine if vectors $\vec{u} = (2, -1, 3)$ and $\vec{v} = (1, 5, 1)$ are orthogonal.

Solution: $\vec{u} \cdot \vec{v} = (2)(1) + (-1)(5) + (3)(1) = 2 - 5 + 3 = 0$. Since the dot product is 0, the vectors are orthogonal.

Problem 4: Calculate the dot product of $\vec{r} = (7, -3, 2)$ and $\vec{s} = (-1, 4, -2)$.

Solution: $\vec{r} \cdot \vec{s} = (7)(-1) + (-3)(4) + (2)(-2) = -7 - 12 - 4 = -23$.

Problem 5: Find the dot product of $\vec{m} = (0, 4, -6)$ and $\vec{n} = (5, -2, 0)$.

Solution: $\vec{m} \cdot \vec{n} = (0)(5) + (4)(-2) + (-6)(0) = 0 - 8 + 0 = -8$.

Problem 6: Are $\vec{x} = (4, 2, -1)$ and $\vec{y} = (-2, 4, 0)$ orthogonal?

Solution: $\vec{x} \cdot \vec{y} = (4)(-2) + (2)(4) + (-1)(0) = -8 + 8 + 0 = 0$. Yes, they are orthogonal.

Problem 7: Compute the dot product of $\vec{a} = (-3, -2, -1)$ and $\vec{b} = (2, 3, 4)$.

Solution: $\vec{a} \cdot \vec{b} = (-3)(2) + (-2)(3) + (-1)(4) = -6 - 6 - 4 = -16$.

🔑 Conclusion

The dot product of 3D vectors is a fundamental concept with widespread applications. By understanding its definition, geometric interpretation, and properties, you can solve various problems in mathematics, physics, and computer science. Keep practicing, and you'll master it in no time!