Linear combination of vectors

A simple Geogebra worksheet


The black vector is the sum of the red and blue dashed vectors (you need to move the sliders \( a \) and \( b \) to make the dashed vectors appear). Can you make the black vector equal to the green vector? You may get lucky with trial and error, but it can be frustrating – it’s far better to set up a system of equations to determine what the values of \( a \) and \( b \) should be.

You can change the red, blue, and green vectors either by clicking-and-dragging the points on the graph, or by double-clicking on the points B, C, D in the left-hand column and editing the values. While this worksheet might not help you find the coefficients in a specific linear combination, it’s designed to help you visualize, and you can use it to check your work for most of the two-dimensional linear combination problems we’ll see.