Chapter 1. Vectors in two dimensions FREE
1.1 Scalars and vectors
In Technical Sciences we distinguish between scalars and vectors. A scalar has only magnitude, for example mass, time and temperature. A vector has magnitude and direction, for example displacement, velocity, acceleration and force. We draw a vector as an arrow: the length of the arrow represents the magnitude and the arrowhead indicates the direction.
1.2 Vectors in two dimensions
Many real vectors do not lie neatly along one axis. A force can act at an angle, or an object can move east and north at the same time. Such vectors lie in a plane and have two dimensions. We analyse them by breaking each slanted vector into two perpendicular parts: a horizontal component and a vertical component.
1.3 Components of a vector
If a vector with magnitude \( F \) makes an angle \( \theta \) with the horizontal, then the two components are:
- Horizontal component: \( F_x = F\cos\theta \)
- Vertical component: \( F_y = F\sin\theta \)
Example Calculate the components of a force of \( 20 \) N that makes an angle of \( 30^\circ \) with the horizontal. The horizontal component is \( F_x = 20\cos 30^\circ = 17{,}3 \) N and the vertical component is \( F_y = 20\sin 30^\circ = 10 \) N. The two components together have exactly the same effect as the original slanted force.
1.4 Addition of vectors
When two or more vectors act together, the single vector with the same effect as all the vectors together is the resultant. For vectors along the same line we simply add or subtract: in the same direction the magnitudes add, in opposite directions they subtract.
For vectors that are perpendicular to each other, we use the theorem of Pythagoras. If \( F_x \) and \( F_y \) are perpendicular, then the magnitude of the resultant is:
\( R = \sqrt{F_x^2 + F_y^2} \)
and the direction is given by \( \tan\theta = \dfrac{F_y}{F_x} \).
Example An object is pulled by a force of \( 3 \) N to the east and a force of \( 4 \) N to the north. The magnitude of the resultant is \( R = \sqrt{3^2 + 4^2} = \sqrt{25} = 5 \) N. The direction is \( \theta = \tan^{-1}\left(\dfrac{4}{3}\right) = 53^\circ \) north of east.
1.5 Adding vectors graphically
We can also determine a resultant graphically with the head-to-tail method: draw the first vector to scale, place the tail of the second vector against the head of the first, and continue in this way. The resultant is the arrow that runs from the very first tail to the very last head. This method works well when there are more than two vectors, but for precise answers the calculation with components remains the most reliable.
1.6 Why it matters
Engineers use vector analysis daily: to calculate the net force on a bridge's supports, to determine the tension in the cables of a crane, and to predict the motion of vehicles. By dividing each slanted vector into horizontal and vertical components, we can turn complicated problems into simple additions. Always remember: an answer for a vector is only complete when both the magnitude and the direction are given.