A man drove his car northward,then turned left and drove 10 km and again turned left and drove 20 km.He found himself 10 km west of his starting point.How far did he drive northward initially.

make a simple sketch , to see that he completed a rectangle, so he must have traveled 20 km north

or, suppose you don't see that right away:

label his trips, AB, then BC, then BD
from D draw a horizontal to meet AB at E
so in right-angled triangle AED,
AE^2 + 10^2 = 10^2
AE = 0

To find out how far the man drove northward initially, we need to apply the concept of vector addition.

Let's break down the movement of the car into two components: northward and westward.

1. The man initially drove northward for a certain distance, let's call it 'x' km. So, the northward component of the car's movement is x km.

2. The man then turned left and drove 10 km. This means the westward component of the car's movement is 10 km.

3. Finally, the man turned left again and drove 20 km. This means the westward component of the car's movement increased to 10 km + 20 km = 30 km.

Since the man found himself 10 km west of his starting point, the westward component of the car's movement is 10 km.

Using vector addition, we can equate the northward component of the car's movement (x km) with the eastward component (-30 km) and the westward component (10 km).

x km + (-30 km) + 10 km = 0 km

Simplifying the equation, we get:

x km - 20 km = 0 km

Adding 20 km to both sides of the equation:

x km = 20 km

Therefore, the man initially drove northward for 20 km.