Question

Show that points \(A, B,\) and \(C\) are all on one line if and only if \(\overrightarrow{A B} \times \overrightarrow{A C}=0\).

Short answer

Points are collinear if the cross product of vectors \(\overrightarrow{A B}\) and \(\overrightarrow{A C}\) is zero.

Step-by-step solution

Understanding Collinearity

To determine if points \(A, B,\) and \(C\) are on a straight line, they must be collinear. Collinearity of points can also be determined using vectors. If points are collinear, the vectors formed by these points should be parallel.

Defining Vectors

Let's define vectors \(\overrightarrow{A B}\) and \(\overrightarrow{A C}\) using the points \(A, B,\) and \(C\). The vectors are formed as follows: \(\overrightarrow{AB} = B - A\) and \(\overrightarrow{AC} = C - A\).

Using the Cross Product

The cross product of two vectors, \(\overrightarrow{A B} \times \overrightarrow{A C}\), checks if they are parallel. If the vectors are parallel, the result of the cross product will be zero: \(\overrightarrow{A B} \times \overrightarrow{A C} = 0\). This indicates that the magnitude of the parallelogram formed by the vectors is zero, a condition met only when the vectors are parallel.

Interpreting the Result

If \(\overrightarrow{A B} \times \overrightarrow{A C} = 0\), it implies that vectors \(\overrightarrow{AB}\) and \(\overrightarrow{AC}\) are parallel. Consequently, if \(\overrightarrow{A B} \times \overrightarrow{A C} = 0\), points \(A, B,\) and \(C\) are collinear. Conversely, if the points are truly on a line (collinear), then all related vectors should satisfy this cross product condition being zero.

Key concepts

Understanding Vectors

Vectors are fundamental in mathematics and physics. They have both magnitude and direction, differentiating them from simple numbers. Imagine arrows pointing from one spot to another; these are vectors. Their length shows the magnitude, and the pointing direction shows where they head. When dealing with points like A, B, and C, we define vectors connecting these points.

For instance, vector \(\overrightarrow{AB}\) is the vector from point A to point B. We calculate it by subtracting the coordinates of A from B. Similarly, vector \(\overrightarrow{AC}\) extends from A to C. These vectors help determine various relationships between points, such as whether they align in a straight line, which we'll explore using other tools like the cross product.

Exploring the Cross Product

The cross product is a crucial operation in vector mathematics. It takes two vectors and results in a third vector perpendicular to both. For vectors \(\overrightarrow{AB}\) and \(\overrightarrow{AC}\), this gives \(\overrightarrow{AB} \times \overrightarrow{AC}\). A special property of the cross product is its magnitude. It tells us the area of the parallelogram formed by the two vectors.

If the result is zero, the vectors do not form a parallelogram. Instead, they lie on the same line, indicating parallelism. This is a key indicator of collinearity. So when \(\overrightarrow{AB} \times \overrightarrow{AC} = 0\), vectors \(\overrightarrow{AB}\) and \(\overrightarrow{AC}\) are parallel, proving the points A, B, and C are collinear.

Understanding Parallelism

Parallelism between two vectors means they share the same direction, albeit they might have different magnitudes. In practical terms, this translates to two vectors lying along the same line. For the vectors \(\overrightarrow{AB}\) and \(\overrightarrow{AC}\) to be parallel, their cross product must be zero.

This example highlights how vectors relate geometrically. Two vectors are parallel when they represent the same or opposite direction, showing no width in the two-dimensional space between them. This absence of deviation ensures that if these vectors are attached to a common starting point, the destination points must lie straight. Thus, understanding parallelism is key to seeing why the zero cross product condition leads to collinearity of the points A, B, and C.