Given two points A (1, 2, 3) and B (4, -1, 5), find the position vector of B in respect to A.
Answer
Subtracting \(\vec{A}\) from \(\vec{B}\), we get \(\vec{BA} = \vec{B} - \vec{A} = \langle 4, -1, 5 \rangle - \langle 1, 2, 3 \rangle = \langle 4-1, -1-2, 5-3 \rangle = \langle 3, -3, 2 \rangle\).
Step 1 ：The position vector of B in respect to A, denoted as \(\vec{BA}\), is given by \(\vec{B} - \vec{A}\).
Step 2 ：First, we find the position vectors of A and B. The position vector of a point \(P (x, y, z)\) is given by \(\vec{OP} = \langle x, y, z \rangle\), where O is the origin (0, 0, 0). Therefore, \(\vec{A} = \langle 1, 2, 3 \rangle\) and \(\vec{B} = \langle 4, -1, 5 \rangle\).
Step 3 ：Subtracting \(\vec{A}\) from \(\vec{B}\), we get \(\vec{BA} = \vec{B} - \vec{A} = \langle 4, -1, 5 \rangle - \langle 1, 2, 3 \rangle = \langle 4-1, -1-2, 5-3 \rangle = \langle 3, -3, 2 \rangle\).
