Rina's 30 m east and 40 m north walk came out at 50 m on a ruler. But a scale drawing takes minutes and a desk. And the answer is only as good as the ruler.
When the two vectors are perpendicular, there is a faster way that gives an exact answer.
Rina's two walks met at a right angle. So her drawing contained a right triangle. The two walks are its two short sides. The start-to-finish line is its longest side.
Math class has an exact rule for right triangles: the Pythagorean theorem. For two perpendicular vectors, the theorem gives the resultant's magnitude directly.
Square each magnitude, add the squares, and take the square root of the sum. The answer is exact. No ruler is needed.
Rina walks 30 m east, then 40 m north. Calculate the magnitude of her resultant displacement.
Write down the values in the question:
A = 30 m, B = 40 m, R = ?
Write down the equation:
R = √(A² + B²)
Substitute in the values:
R = √(30² + 40²) = √(900 + 1600) = √2500
R = 50 m: the same answer the ruler gave, now exact.
Two perpendicular velocities act together on a boat: 6 m/s east and 8 m/s north. Calculate the magnitude of the resultant velocity.
Write down the values in the question:
A = 6 m/s, B = 8 m/s, R = ?
Write down the equation:
R = √(A² + B²)
Substitute in the values:
R = √(6² + 8²) = √(36 + 64) = √100
R = 10 m/s.
Now make the move quick: square, add, root, five times in a row. A sixth item then checks the boundary: when the shortcut is allowed at all.
Two perpendicular displacements: 9 m east and 12 m north. Enter the resultant's magnitude in m.
Two perpendicular displacements: 7 m south and 24 m east. Enter the resultant's magnitude in m.
Two perpendicular velocities: 16 m/s east and 30 m/s south. Enter the resultant's magnitude in m/s.
Two perpendicular displacements: 33 m west and 44 m south. Enter the resultant's magnitude in m.
Two perpendicular velocities: 10 m/s north and 24 m/s east. Enter the resultant's magnitude in m/s.
Two velocities meet at 50°. Can R = √(A² + B²) give their resultant's magnitude?
Two perpendicular displacements act on a cart: 60 m east and 11 m north. Enter the resultant's magnitude in m.
A drone flies 33 m west, then 56 m north. Enter its straight-line distance from the start in m.
Two perpendicular velocities: 20 m/s west and 21 m/s north. Enter the resultant's magnitude in m/s.