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 \(\overrightarrow{AB} \times \overrightarrow{AC} = 0\).

Step-by-step solution

Understanding Vectors Between Points

Consider points \(A, B,\) and \(C\) in space. The vectors between these points are \(\overrightarrow{AB} = \mathbf{B} - \mathbf{A}\) and \(\overrightarrow{AC} = \mathbf{C} - \mathbf{A}\), where \(\mathbf{A}, \mathbf{B}, \mathbf{C}\) are the position vectors.

Defining Collinearity

Points are collinear if they lie on the same straight line. Vectorially, points \(A, B,\) and \(C\) are collinear if one vector is a scalar multiple of the other, meaning \(\overrightarrow{AB} = k \cdot \overrightarrow{AC}\) for some scalar \(k\).

Cross Product and Collinearity

The cross product \(\overrightarrow{AB} \times \overrightarrow{AC}\) is used to check if vectors are parallel. If \(\overrightarrow{AB}\) and \(\overrightarrow{AC}\) are parallel, their cross product will be the zero vector: \(\overrightarrow{AB} \times \overrightarrow{AC} = \mathbf{0}\).

Implication of Zero Cross Product

If \(\overrightarrow{AB} \times \overrightarrow{AC} = \mathbf{0}\), the vectors are parallel, indicating that points \(A, B,\) and \(C\) are aligned on the same line.

Conclusion

Therefore, points \(A, B,\) and \(C\) are collinear if and only if \(\overrightarrow{AB} \times \overrightarrow{AC} = \mathbf{0}\), confirming the statement.

Key concepts

Collinearity

Collinearity is a term used in geometry to describe a situation where three or more points lie on a single straight line. This concept is central in vector mathematics, as it connects algebraic operations with geometrical interpretations. In the context of vectors, points are considered collinear if the line connecting the first two points continues to the third point. Collinearity can be mathematically verified by checking if one of the vectors is a scalar multiple of the other. This is expressed as \( \overrightarrow{AB} = k \cdot \overrightarrow{AC} \), where \( k \) is a constant scalar. If such a scalar exists, it confirms that the points do not deviate from a straight path when moving from one to the next.
This property makes collinearity a valuable tool for checking alignment without needing to graphically visualize the points.

Cross Product

Understanding the cross product is crucial when dealing with vectors in geometry. The cross product of two vectors in space gives a third vector that is perpendicular to the plane containing the original vectors.
The cross product is evaluated using the determinants of a matrix constructed with the vectors' components. For vectors \( \overrightarrow{AB} = (a_1, a_2, a_3) \) and \( \overrightarrow{AC} = (b_1, b_2, b_3) \), their cross product is given by:

• \( \overrightarrow{AB} \times \overrightarrow{AC} = \begin{vmatrix}\mathbf{i} & \mathbf{j} & \mathbf{k} \ a_1 & a_2 & a_3 \ b_1 & b_2 & b_3 \end{vmatrix} \)

This vector operation is critical as it helps determine parallelism between vectors. When the cross product results in the zero vector, \( \mathbf{0} \), the given vectors are parallel, indicating that the points they connect are collinear. This confirmation by the zero cross product can greatly assist in resolving questions of alignment.

Vectors and Geometry

Vectors offer powerful ways to interpret and solve geometric problems. In geometry, vectors are portrayed as arrows in space that define direction and magnitude. They can represent points, lines, and even surfaces.
When applied to points like \(A, B,\) and \(C\), vectors help streamline calculations and visualizations. The position vectors \(\mathbf{A}, \mathbf{B}, \mathbf{C}\) denote the location of these points. The difference of position vectors gives us directional vectors, such as \( \overrightarrow{AB} = \mathbf{B} - \mathbf{A} \).
A common geometric problem is determining the linearity of points, for which vectors are particularly useful. Operations like taking the cross product of vectors help to assert whether points form part of a single line by confirming their parallel nature. Thus, vectors bridge the conceptual gap between numerical data and spatial orientation.