Find an initial point P of a nonzero vector 𝐮=P Q with terminal point Q(3,0,-5) and such that (a

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Find an initial point P of a nonzero vector 𝐮=P Q with...

Find an initial point $P$ of a nonzero vector $\mathbf{u}=\overrightarrow{P Q}$ with terminal point $Q(3,0,-5)$ and such that
(a) u has the same direction as $v=(4,-2,-1)$.
(b) $\mathbf{u}$ is oppositely directed to $\mathbf{v}=(4,-2,-1)$.

Key Concepts

Vector Representation Using Points

This concept explains how a vector in coordinate geometry is defined by its initial and terminal points. The vector is expressed as the difference in coordinates between the terminal point and the initial point, which is a fundamental idea for understanding vector displacement.

Scalar Multiplication of Vectors

Scalar multiplication involves multiplying a vector by a real number, which scales the vector's magnitude without changing its direction when the scalar is positive and reverses the direction when the scalar is negative. This property is crucial for adjusting vector length and orientation.

Parallel and Anti-Parallel Vectors

Two vectors are parallel if one is a scalar multiple of the other. When the scalar is positive, the vectors have the same direction, and when it is negative, the vectors are oppositely directed (anti-parallel). This concept underlies the method of aligning vectors along the same or reverse orientations.

Determining Initial or Terminal Points Based on Vector Direction

This idea involves using the known terminal point and a given directional vector to find a suitable initial point such that the resulting vector has the required direction. By exploiting the relationship between the vector's endpoints and its scalar multiple relation to the direction vector, one can adjust the position of the initial point accordingly.

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