Welcome to Professor Baker's Math Class! - April 18th: Sections 12-1 & 12-2

Hello everyone! Today, we'll be reviewing the key concepts covered in class on April 18th, focusing on sections 12-1 and 12-2. Get ready to explore the exciting world of three-dimensional coordinate systems and vectors!

Three-Dimensional Coordinate Systems

First, we introduced the three-dimensional coordinate system. Remember that it's defined by three mutually perpendicular axes: the x-axis, the y-axis, and the z-axis. A point in 3D space is represented by an ordered triple $(x, y, z)$. We visualized the coordinate planes (xy-plane, xz-plane, yz-plane) which help divide the space into octants.

For example, consider the equations $x^2 + y^2 = 1$ and $z = 3$. The points satisfying these equations form a circle of radius 1 in the plane $z=3$. Furthermore, the equation $x^2 + y^2 = 1$ alone represents a cylinder in $R^3$.

Distance Formula in Three Dimensions

A crucial concept we covered was the distance formula in three dimensions. If we have two points, $P_1(x_1, y_1, z_1)$ and $P_2(x_2, y_2, z_2)$, the distance between them, denoted as $|P_1P_2|$, is given by:

$$|P_1P_2| = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}$$

Example: Find the distance between the points $P(2, -1, 7)$ and $Q(1, 3, 5)$.

Using the formula:

$$|PQ| = \sqrt{(1 - 2)^2 + (3 - (-1))^2 + (5 - 7)^2} = \sqrt{(-1)^2 + (4)^2 + (-2)^2} = \sqrt{1 + 16 + 4} = \sqrt{21}$$

Vectors

We then moved on to vectors. Remember, a vector is a quantity that has both magnitude and direction. We can represent a vector in 3D space as $\mathbf{a} = \langle a_1, a_2, a_3 \rangle$, where $a_1$, $a_2$, and $a_3$ are the components of the vector along the x, y, and z axes, respectively.

Vector Operations:

  • Vector Addition: If $\mathbf{u} = \langle u_1, u_2, u_3 \rangle$ and $\mathbf{v} = \langle v_1, v_2, v_3 \rangle$, then $\mathbf{u} + \mathbf{v} = \langle u_1 + v_1, u_2 + v_2, u_3 + v_3 \rangle$.
  • Scalar Multiplication: If $c$ is a scalar and $\mathbf{v} = \langle v_1, v_2, v_3 \rangle$, then $c\mathbf{v} = \langle cv_1, cv_2, cv_3 \rangle$.

The magnitude of a vector $\mathbf{a} = \langle a_1, a_2, a_3 \rangle$, denoted as $|\mathbf{a}|$, is given by:

$$|\mathbf{a}| = \sqrt{a_1^2 + a_2^2 + a_3^2}$$

A unit vector is a vector whose length is 1. To find a unit vector $\mathbf{u}$ in the direction of a vector $\mathbf{a}$, we use the formula:

$$ \mathbf{u} = \frac{1}{|\mathbf{a}|} \mathbf{a} $$

Example: If $\mathbf{a} = \langle 4, 0, 3 \rangle$ and $\mathbf{b} = \langle -2, 1, 5 \rangle$, then:

  • $\mathbf{a} + \mathbf{b} = \langle 2, 1, 8 \rangle$
  • $\mathbf{a} - \mathbf{b} = \langle 6, -1, -2 \rangle$
  • $3\mathbf{b} = \langle -6, 3, 15 \rangle$
  • $2\mathbf{a} + 5\mathbf{b} = \langle -2, 5, 31 \rangle$
  • $|\mathbf{a}| = \sqrt{4^2 + 0^2 + 3^2} = 5$
  • $|\mathbf{b}| = \sqrt{(-2)^2 + 1^2 + 5^2} = \sqrt{30}$

These sections laid the groundwork for understanding spatial relationships and vector operations, which are fundamental in various fields like physics, engineering, and computer graphics. Keep practicing, and you'll master these concepts in no time!