A commentator says that the fielder is thirty metres from the batsman. That is not enough to place the fielder. Thirty metres in which direction? On a cricket ground, one distance never fixes a position. Two numbers are needed, or a distance and a direction. The last chapter needed only one number because the motion was along a line. A plane needs two.
Two axes, two numbers
Choose an origin O on the plane. Draw two perpendicular axes through O, the \(x\) axis and the \(y\) axis. Any point in the plane is then fixed by two numbers, its \(x\) coordinate and its \(y\) coordinate. The point \((3, 4)\,\text{m}\) lies \(3\,\text{m}\) along the \(x\) axis and \(4\,\text{m}\) along the \(y\) axis from O. The two numbers together are the position of the point.
The position can also be written as one arrow. The position vector \(\vec r\) is the arrow from the origin O to the point. Its length is the distance of the point from O. Its direction is the direction of the point as seen from O. In terms of the unit vectors \(\hat i\) along \(x\) and \(\hat j\) along \(y\),
\[ \vec r = x\,\hat i + y\,\hat j, \qquad |\vec r| = \sqrt{x^2 + y^2} \]
For the point \((3, 4)\,\text{m}\), \(\vec r = 3\hat i + 4\hat j\) and \(|\vec r| = \sqrt{9 + 16} = 5\,\text{m}\).
The position vector of a body is the vector from the origin of the frame of reference to the position of the body. In a plane, \(\vec r = x\,\hat i + y\,\hat j\), where \(x\) and \(y\) are the coordinates of the body.
Displacement is a vector in the plane
Suppose the body moves from position \(\vec r_1 = x_1\hat i + y_1\hat j\) to position \(\vec r_2 = x_2\hat i + y_2\hat j\). The displacement is the change in the position vector.
\[ \Delta\vec r = \vec r_2 - \vec r_1 = (x_2 - x_1)\,\hat i + (y_2 - y_1)\,\hat j \]
Displacement is found by subtracting the coordinates one axis at a time. The \(x\) component of the displacement is the change in \(x\). The \(y\) component is the change in \(y\). The path between the two positions does not enter at all. As in one dimension, displacement depends only on the end points.
A fielder starts at the point \((10, 5)\,\text{m}\) and, after running along a curve to collect the ball, ends at \((22, 14)\,\text{m}\). Find the displacement vector and its magnitude.
\(\Delta\vec r = (22 - 10)\,\hat i + (14 - 5)\,\hat j = 12\hat i + 9\hat j\) metres.
The magnitude is \(\sqrt{12^2 + 9^2} = \sqrt{144 + 81} = \sqrt{225} = 15\,\text{m}\).
The fielder may have run \(20\,\text{m}\) or \(30\,\text{m}\) along the curve; that distance does not affect the displacement.
A body moves from the point \((2, 1)\,\text{m}\) to the point \((5, 5)\,\text{m}\). Find the magnitude of its displacement in metres.
The magnitude of the displacement is not \(\Delta x + \Delta y\). The two components are perpendicular, so they add like the two sides of a right triangle, by Pythagoras. \(3\,\text{m}\) east and \(4\,\text{m}\) north give \(5\,\text{m}\), not \(7\,\text{m}\).
A runner completes one full lap of a \(400\,\text{m}\) athletics track and returns to the starting line. Which statement is correct?
The displacement does not depend on where the origin is placed. Move the origin, and both \(\vec r_1\) and \(\vec r_2\) change by the same vector, so their difference \(\Delta\vec r\) does not change. This is why displacement, not position, is the quantity that velocity is built from.
Why does a position in a plane need two numbers, while a position on a straight line needs only one?
On a straight line every point is fixed by its signed distance from the origin, one number. In a plane a point can be moved sideways without changing its distance along the first axis, so a second number, the coordinate along a second axis, is needed to fix it. Two perpendicular axes give two independent numbers, and together they fix the point.
- Adding the components arithmetically to get the magnitude of a vector.Components along perpendicular axes combine by Pythagoras: \(|\vec r| = \sqrt{x^2 + y^2}\).
- Confusing the position vector with the displacement vector.Position is measured from the origin. Displacement is measured from the previous position and does not depend on the origin.