Vectors are fundamental in 3D mathematics, particularly in fields like computer graphics, physics, and engineering. This section will cover the basics of vectors in three-dimensional space, including their representation, operations, and applications.

• Definition: A vector in 3D space is an ordered triplet of numbers, representing a point or direction in space.
• Notation: Vectors are typically denoted by bold letters or with an arrow above the letter, e.g., v or \(\vec{v}\).
• Components: A vector v can be represented as \(\vec{v} = (v_x, v_y, v_z)\), where \(v_x\), \(v_y\), and \(v_z\) are the components along the x, y, and z axes, respectively.

• Addition: The sum of two vectors \(\vec{u} = (u_x, u_y, u_z)\) and \(\vec{v} = (v_x, v_y, v_z)\) is given by: \[ \vec{u} + \vec{v} = (u_x + v_x, u_y + v_y, u_z + v_z) \]
• Subtraction: The difference between two vectors \(\vec{u}\) and \(\vec{v}\) is: \[ \vec{u} - \vec{v} = (u_x - v_x, u_y - v_y, u_z - v_z) \]
• Scalar Multiplication: Multiplying a vector \(\vec{v}\) by a scalar \(k\) results in: \[ k\vec{v} = (kv_x, kv_y, kv_z) \]
• Magnitude (Length): The magnitude of a vector \(\vec{v}\) is: \[ |\vec{v}| = \sqrt{v_x^2 + v_y^2 + v_z^2} \]
• Normalization: A unit vector in the direction of \(\vec{v}\) is obtained by dividing \(\vec{v}\) by its magnitude: \[ \hat{v} = \frac{\vec{v}}{|\vec{v}|} \]

Example 1: Vector Addition

Given \(\vec{a} = (1, 2, 3)\) and \(\vec{b} = (4, 5, 6)\), find \(\vec{a} + \vec{b}\).

Solution: \[ \vec{a} + \vec{b} = (1 + 4, 2 + 5, 3 + 6) = (5, 7, 9) \]

Example 2: Magnitude of a Vector

Find the magnitude of \(\vec{c} = (3, -4, 12)\).

Solution: \[ |\vec{c}| = \sqrt{3^2 + (-4)^2 + 12^2} = \sqrt{9 + 16 + 144} = \sqrt{169} = 13 \]

Given \(\vec{u} = (7, 8, 9)\) and \(\vec{v} = (2, 3, 4)\), find \(\vec{u} - \vec{v}\).

Solution: \[ \vec{u} - \vec{v} = (7 - 2, 8 - 3, 9 - 4) = (5, 5, 5) \]

Multiply the vector \(\vec{w} = (1, -2, 3)\) by the scalar \(k = 4\).

Solution: \[ 4\vec{w} = 4(1, -2, 3) = (4, -8, 12) \]

Normalize the vector \(\vec{d} = (6, 8, 0)\).

Solution: First, find the magnitude of \(\vec{d}\): \[ |\vec{d}| = \sqrt{6^2 + 8^2 + 0^2} = \sqrt{36 + 64} = \sqrt{100} = 10 \] Then, divide each component by the magnitude: \[ \hat{d} = \frac{\vec{d}}{|\vec{d}|} = \frac{(6, 8, 0)}{10} = (0.6, 0.8, 0) \]

• Mistake: Forgetting to apply the square root when calculating the magnitude.
  • Tip: Always double-check the formula for magnitude to ensure all components are squared and summed before taking the square root.
• Mistake: Mixing up vector addition and subtraction.
  • Tip: Carefully follow the signs and ensure each component is correctly added or subtracted.

In this section, we covered the basics of vectors in 3D space, including their representation, fundamental operations, and practical examples. Understanding these concepts is crucial for manipulating and interpreting 3D graphics and physical simulations. In the next section, we will delve into the dot product and cross product, which are essential for further vector operations and applications in 3D space.