Understanding Collinear Points in 3D Geometry

When three or more points lie on the same straight line, they are said to be collinear points. In three-dimensional space, if points A, B, and C are collinear, the vector formed by any two points, say \(\vec{AB}\), will be parallel to the vector formed by another pair of points, say \(\vec{BC}\). This parallelism can be expressed mathematically by stating that one vector is a scalar multiple of the other, i.e., \(\vec{AB} = k \vec{BC}\) for some scalar \(k\).

We are given three points: A(x, y, -3), B(2, 0, -1), and C(4, 2, 3). Since these points lie on a straight line, they are collinear. We can use the vector condition for collinearity to find the unknown values of x and y.

Step-by-Step Solution to Find x and y

Step 1: Formulate Vectors between the Points

Let's form the vectors \(\vec{AB}\) and \(\vec{BC}\) using the given coordinates. The vector from point \((x_1, y_1, z_1)\) to point \((x_2, y_2, z_2)\) is given by \((x_2-x_1, y_2-y_1, z_2-z_1)\).

• Vector \(\vec{AB}\): From A(x, y, -3) to B(2, 0, -1)
• \(\vec{AB} = (2-x, 0-y, -1 - (-3))\)
• \(\vec{AB} = (2-x, -y, -1 + 3)\)
• \(\vec{AB} = (2-x, -y, 2)\)

• Vector \(\vec{BC}\): From B(2, 0, -1) to C(4, 2, 3)
• \(\vec{BC} = (4-2, 2-0, 3 - (-1))\)
• \(\vec{BC} = (2, 2, 3 + 1)\)
• \(\vec{BC} = (2, 2, 4)\)

Step 2: Apply the Collinearity Condition

Since the points A, B, and C are collinear, the vectors \(\vec{AB}\) and \(\vec{BC}\) must be parallel. This means \(\vec{AB} = k \vec{BC}\) for some scalar \(k\).

Substituting the component forms of the vectors: \((2-x, -y, 2) = k(2, 2, 4)\)

This vector equation can be broken down into three separate scalar equations by equating the corresponding components:

1. \(2-x = 2k\)
2. \(-y = 2k\)
3. \(2 = 4k\)

Step 3: Solve the System of Equations

We have a system of three equations with three unknowns (x, y, and k). We can solve for k first using the third equation, as it only involves k.

From equation (3):

\(2 = 4k\)

\(k = \frac{2}{4} = \frac{1}{2}\)

Now substitute the value of \(k = \frac{1}{2}\) into equations (1) and (2) to find x and y.

Substitute k into equation (1):

\(2-x = 2k\)

\(2-x = 2 \times \frac{1}{2}\)

\(2-x = 1\)

\(x = 2 - 1\)

\(x = 1\)

Substitute k into equation (2):

\(-y = 2k\)

\(-y = 2 \times \frac{1}{2}\)

\(-y = 1\)

\(y = -1\)

Step 4: State the Values of x and y

From the calculations, we found that \(x = 1\) and \(y = -1\). The values of x and y are 1 and -1 respectively.

Let's verify this by plugging \(x=1\) and \(y=-1\) back into the vectors.

Point A becomes (1, -1, -3).

• \(\vec{AB} = (2-1, 0-(-1), -1 - (-3)) = (1, 1, 2)\)
• \(\vec{BC} = (2, 2, 4)\)

Is \(\vec{AB}\) a scalar multiple of \(\vec{BC}\)? Yes, \(\vec{BC} = 2 \times (1, 1, 2) = 2 \vec{AB}\). So \(\vec{AB} = \frac{1}{2} \vec{BC}\), which matches our value of \(k=\frac{1}{2}\). The points are indeed collinear for \(x=1\) and \(y=-1\).

The values of x and y are 1 and -1 respectively.

Revision Table: Key Concepts

Additional Information on Collinearity

Besides using vectors, there are other ways to check if three points A, B, and C are collinear in 3D:

• Distance Formula: If the sum of the distances between two pairs of points equals the distance between the third pair, the points are collinear. For example, if \(AB + BC = AC\), the points are collinear. Calculating distances in 3D involves the formula \(\sqrt{(x_2-x_1)^2 + (y_2-y_1)^2 + (z_2-z_1)^2}\).
• Area of Triangle: The area of the triangle formed by three collinear points is zero. In 3D, the area of triangle ABC can be found using the magnitude of the cross product of two vectors forming the sides, e.g., Area \(= \frac{1}{2} |\vec{AB} \times \vec{AC}|\). If the points are collinear, \(|\vec{AB} \times \vec{AC}| = 0\).

The vector method used in the solution is generally the most straightforward when dealing with unknown coordinates, as it directly leads to a system of linear equations. Understanding collinearity is fundamental in coordinate geometry and vector algebra, used in various applications including physics and computer graphics.