Operations with Vectors: Apply It

  • Perform vector addition and scalar multiplication.
  • Perform operations with vectors in terms of i and j .
  • Find the dot product of two vectors.

Vectors and Bearing

When adding vectors, we add corresponding components. The sum [latex]\mathbf{u} + \mathbf{v} = \langle u_1, u_2 \rangle + \langle v_1, v_2 \rangle = \langle u_1 + v_1, u_2 + v_2 \rangle[/latex] represents the combined effect of both vectors.

An airplane is flying at an airspeed of 200 miles per hour headed on a SE bearing of 140°. A north wind (from north to south) is blowing at 16.2 miles per hour. What are the ground speed and actual bearing of the plane?Image of a plan flying SE at 140 degrees and the north wind blowing.

In navigation, bearings are measured clockwise from north. To convert to standard position (counterclockwise from east), use the relationship: east component = [latex]|\mathbf{v}|\sin(\text{bearing})[/latex] and north component = [latex]|\mathbf{v}|\cos(\text{bearing})[/latex].