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Vector Algebra

Question
CBSEENMA12033956

ABC is a triangle and D is the mid-point of the side BC. Show that AB with rightwards arrow on top space plus space AC with rightwards arrow on top space equals space 2 space AD with rightwards arrow on top.

Solution

In increment ABD comma space space space space space space space space space space space space space space space space space space AB with rightwards arrow on top space equals space AD with rightwards arrow on top space plus space BD with rightwards arrow on top                                    ...(1)
In  increment space ADC comma space space space space space space AC with rightwards arrow on top space equals space AD with rightwards arrow on top space plus space DC with rightwards arrow on top                                            ...(2)
Adding (1) and (2),  we get

     AB with rightwards arrow on top space plus space AC with rightwards arrow on top space equals space 2 space AD with rightwards arrow on top space plus space open parentheses DB with rightwards arrow on top space plus space DC with rightwards arrow on top close parentheses
                                               equals space 2 space AD with rightwards arrow on top space plus space DB with rightwards arrow on top space space space space space space space space space space space space space space space space space space space space space space open square brackets because space space straight D space is space mid minus point space of space BC comma space space space space space because space space BD with rightwards arrow on top space equals DC with rightwards arrow on top close square brackets
equals space 2 space stack AD space with rightwards arrow on top space minus space BD with rightwards arrow on top space plus space BD with rightwards arrow on top
therefore     AB with rightwards arrow on top space plus space AC with rightwards arrow on top space equals 2 space space AD with rightwards arrow on top