This tutorial presents vector concepts including components, magnitude, scalar multiplication, and vector addition, with worked examples and detailed explanations.
Vectors are mathematical quantities that have both a magnitude (size) and a direction. They are commonly used to represent physical quantities such as force, velocity, and displacement.
The figure below shows a vector v with initial point A and terminal point B.
If a vector v has initial point A\( (a_1, a_2) \) and terminal point B\( (b_1, b_2) \), then its component form is obtained by subtracting coordinates:
\[ \vec{v} = \langle b_1 - a_1,\; b_2 - a_2 \rangle \]This tells us how much the vector moves horizontally and vertically.
If a vector is given in component form \(\vec{v} = \langle v_1, v_2 \rangle\), then its magnitude (or length) is found using the Pythagorean theorem:
\[ \|\vec{v}\| = \sqrt{v_1^2 + v_2^2} \]Find the components and magnitude of the vector with initial point A\( (2,3) \) and terminal point B\( (4,5) \).
Solution:
First, compute the components:
\[ \vec{v} = \langle 4 - 2,\; 5 - 3 \rangle = \langle 2,\; 2 \rangle \]Now compute the magnitude:
\[ \|\vec{v}\| = \sqrt{2^2 + 2^2} = \sqrt{8} = 2\sqrt{2} \]Multiplying a vector by a real number scales its length. If \(\vec{v} = \langle v_1, v_2 \rangle\) and \(k\) is a real number, then:
\[ k\vec{v} = \langle kv_1,\; kv_2 \rangle \]If \(k > 1\), the vector becomes longer; if \(0 < k < 1\), it becomes shorter; and if \(k < 0\), the direction reverses.
If two vectors are given by \(\vec{v} = \langle v_1, v_2 \rangle\) and \(\vec{u} = \langle u_1, u_2 \rangle\), their sum is obtained by adding corresponding components:
\[ \vec{v} + \vec{u} = \langle v_1 + u_1,\; v_2 + u_2 \rangle \]Let \(\vec{v} = \langle -2, 3 \rangle\) and \(\vec{u} = \langle 4, 6 \rangle\). Find:
Solution:
First compute scalar multiplications, then add vectors.
\[ \vec{v} + 2\vec{u} = \langle -2, 3 \rangle + \langle 8, 12 \rangle = \langle 6, 15 \rangle \] \[ \vec{u} - 4\vec{v} = \langle 4, 6 \rangle + \langle 8, -12 \rangle = \langle 12, -6 \rangle \]Let \(\vec{v} = \langle 1, -2 \rangle\) and \(\vec{u} = \langle u_1, u_2 \rangle\). Find \(u_1\) and \(u_2\) so that:
\[ \vec{v} + 3\vec{u} = \vec{0} \]Solution:
\[ \langle 1 + 3u_1,\; -2 + 3u_2 \rangle = \langle 0, 0 \rangle \]Set each component equal to zero:
\[ 1 + 3u_1 = 0 \quad \Rightarrow \quad u_1 = -\frac{1}{3} \] \[ -2 + 3u_2 = 0 \quad \Rightarrow \quad u_2 = \frac{2}{3} \]1. \(-\vec{v} + 2\vec{u} = \langle -1, -2 \rangle\), \(\vec{v} - \frac{1}{2}\vec{u} = \langle -2, 2 \rangle\)
2. \(u_1 = \frac{8}{3},\quad u_2 = \frac{2}{3}\)