Resolving Vectors in Quadrant I

Any vector can be replaced by two vectors which, acting together, exactly duplicate the effect of the original vector. These replacement vectors are called the components of the vector, and are usually chosen perpendicular (at right angles) to each other. Another name for these perpendicular component vectors is rectangular components. The process of of breaking a 2D vector into its vertical (y) and horizontal (x) components is known as resolving a vector. The first video shows how a vector in the first quadrant is broken down into its rectangular components.

Another way to represent the answer in the video is by writing it in rectangular form:

v = ai + bj   where   a = vx and b = vy

∴ v = 25.3i + 49.3i

Note that i is horizontal component vector of the vector v, and j is vertical component vector. The vector sum ai + bj is called a linear combination of the vector i and j. The magnitude of v = ai + bj is given by:


The process above can also be reversed, where the rectangular components are added up to obtain the resultant vector. This is typically done using the tangent ratio and the Pythagorean theorem. The technique is shown below:

Challenge Question:

Find the resultant of two perpendicular vectors whose magnitudes are 485 and 627. Also find the angle that it makes with the 627-magnitude vector.