Question

The initial and terminal points of a vector \(v\) are given. (a) Sketch the directed line segment, (b) find the component form of the vector, and (c) sketch the vector with its initial point at the origin. Initial point: (2,-1,-2) Terminal point: (-4,3,7)

Short answer

The directed line segment, or vector \(v\), goes from (2,-1,-2) to (-4,3,7). The component form of the vector, \(v\), is (-6,4,9). When plotted from the origin, the vector extends towards the point (-6,4,9).

Step-by-step solution

Sketch the Directed Line Segment

Plot both initial and terminal points in the 3-dimensional space and draw a line from the initial point to the terminal point. The directed line segment or vector \(v\) goes from (2,-1,-2) to (-4,3,7).

Find the Component Form of the Vector

Calculate the x, y, z components of the vector as difference between terminal and initial points. The component form of the vector is: \( x_{component} = -4 - 2 = -6 \), \( y_{component} = 3 - (-1) = 4 \), \( z_{component} = 7 - (-2) = 9 \). Thus, the component form of the vector, \(v\) , is (-6, 4, 9).

Sketch the Vector with Initial Point at the Origin

Plot the point (-6,4,9) in 3-dimensional space starting from the origin and draw a line segment or an arrow from the origin (0,0,0) to (-6,4,9). This shows the vector \(v\) in the position vector form with its initial point at the origin.