Ex 10.3, 13 - Chapter 10 Class 12 Vector Algebra

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Ex 10.3, 13 (Method 1) If , , are unit vectors such that + + = 0 , find the value of . + . + . . Given , , are unit vectors Magnitude of , , is 1 So, = = = 1 Also, + + = 0 So, + + = = 0 Now, | + + |2 = ( + + ) . ( + + ) = . + . + . + . + . + . + . + . + . = . + . + . + . + . + . + . + . + . = . + . + . + 2 . + 2 . + 2 . = . + . + . + 2( . + . + . ) = + + + 2 ( . + . + . ) = 12 + 12 + 12 + 2( . + . + . ) = 1 + 1 + 1 + 2( . + . + . ) = 3 + 2 ( . + . + . ) | + + |2 = 3 + 2 ( . + . + . ) Now, + + = 0 + + 2 = 0 3 + 2 ( . + . + . ) = 0 2( . + . + . ) = 3 ( . + . + . ) = Ex 10.3, 13 (Method 2) If , , are unit vectors such that + + = 0, find the value of . + . + . . Given , , are unit vectors So, = = = 1 Also, ( + + ) = 0 Now, . ( + + ) = . + . + . . 0 = . + . + . 0 = . + . + . 0 = + . + . 0 = 2 + . + . 0 = 2 + . + . 0 = 12 + . + . . + . = 1 Also, . ( + + ) = . + . + . . 0 = . + . + . 0 = . + . + . 0 = . + . + . 0 = . + . + . 0 = . + 2 + . 0 = . + 12 + . . + . = 1 Also . ( + + ) = . + . + . . 0 = . + . + . 0 = . + . + . 0 = . + . + . 0 = . + . + . 0 = . + . + 2 0 = . + . + 12 . + . = 1 Adding (1), (2) and (3), ( . + . ) + ( . + . ) + ( . + . ) = 1 + ( 1) + ( 1) 2 . + 2 . + 2 . = 3 2( . + . + . ) = 3 . + . + . =