Nonzero connecting vectors are parallel exactly when one is a nonzero scalar multiple of the other. Three distinct points are collinear when two such vectors share an endpoint. If AB= d, then AC=k d puts C on the line through A,B. The sign and size of k determine the order of the points. Choose connecting vectors with a common start; subtract positions; factor one vector as a scalar multiple of the other; state the geometric conclusion. If using a non-parallel basis, compare both coefficients. For A(0,0), B(2,1), C(6,3), AC=3 AB, so A,B,C are collinear and AB:BC=1:2. Exam check: parallel vectors based at different points do not establish that all their endpoints lie on one line. Never divide by the zero vector.

Nonzero connecting vectors are parallel exactly when one is a nonzero scalar multiple of the other. Three distinct points are collinear when two such vectors share an endpoint.

If \(\overrightarrow{AB}=\mathbf d\), then \(\overrightarrow{AC}=k\mathbf d\) puts \(C\) on the line through \(A,B\). The sign and size of \(k\) determine the order of the points.

Choose connecting vectors with a common start; subtract positions; factor one vector as a scalar multiple of the other; state the geometric conclusion. If using a non-parallel basis, compare both coefficients.

For \(A(0,0), B(2,1), C(6,3)\), \(\overrightarrow{AC}=3\overrightarrow{AB}\), so \(A,B,C\) are collinear and \(AB:BC=1:2\).

Exam check: parallel vectors based at different points do not establish that all their endpoints lie on one line. Never divide by the zero vector.

Parallelism and collinearity

Parallelism and collinearity