If a → = b → , is it true that a → = ± b → ? Justify your answer.

To prove: a → = ± b → if a → = b → . For two vectors to be equal their corresponding components should be equal. Suppose, a → = 2 i ^ + 3 j ^ & b → = - 3 i ^ + 2 j ^ magnitude of a → a → = 2 2 + 3 2 = 13 Magnitude of b → b → = - 3 2 + 2 2 = 13 Here, magnitudes are same. But 2 i ^ + 3 j ^ ≠ ± ( - 3 i ^ + 2 j ^ ) , because their corresponding components are not equal. Hence, the given statement is not correct.

Find the magnitude of the vector i ^ + j ^ + k ^ .

Magnitude of vector a → = a 1 i ^ + a 2 j ^ + a 3 k ^ is a → = a 1 2 + a 2 2 + a 3 2 . Let given vector be a → = i ^ + j ^ + k ^ So, the magnitude of the vector is given by a → = 1 2 + 1 2 + 1 2 ⇒ a → = 3 Therefore, magnitude of vector a → is 3 .

Find the magnitude of the vector 2 i ^ - 3 j ^ - 7 k ^ .

Let a → = 2 i ^ - 3 j ^ - 7 k ^ . Magnitude of vector a → = a 1 i ^ + a 2 j ^ + a 3 k ^ is a → = a 1 2 + a 2 2 + a 3 2 . So, the magnitude of given vector is a → = 2 2 + - 3 2 + - 7 2 ⇒ a → = 4 + 9 + 49 ⇒ a → = 62 Therefore, magnitude of vector a → is 62 .

Write the scalar components of the vector A B → with initial point A ( 2 , 1 ) and terminal point B - 5 , 7 .

Given that initial point A & final point B of a vector A B → are 2 , 1 and - 5 , 7 respectively. We know that vector, A B → = P . V o f B - P . V o f A Let, O be the origin. Position vector of A , O A → = 2 i ^ + 1 j ^ Position vector of B , O B → = - 5 i ^ + 7 j ^ x 1 = 2 , y 1 = 1 and x 2 = - 5 , y 2 = 7 . So, A B → = O B → - O A → ⇒ A B → = x 2 - x 1 i ^ + y 2 - y 1 j ^ ⇒ A B → = - 5 - 2 i ^ + 7 - 1 j ^ ⇒ A B → = - 7 i ^ + 6 j ^ So, scalar components of the vector A B → are - 7 and 6 .

Find the sum of the vectors a → = i ^ - 2 j ^ + k ^ , b = - 2 i ^ + 4 j ^ + 5 k ^ and c → = i ^ - 6 j ^ - 7 k ^ .

Given vectors: a → = i ^ - 2 j ^ + k ^ , b → = - 2 i ^ + 4 j ^ + 5 k ^ , c → = i ^ - 6 j ^ - 7 k ^ . To find: a → + b → + c → . Here on adding the above vectors a polygon will be formed whose resultant will be lets say r → = O C → . And a → = O A → , b → = A B → , c → = B C → . So, r → = a → + b → + c → ⇒ r → = i ^ - 2 j ^ + k ^ + - 2 i ^ + 4 j ^ + 5 k ^ + i ^ - 6 j ^ - 7 k ^ ⇒ r → = i ^ - 2 j ^ + k ^ - 2 i ^ + 4 j ^ + 5 k ^ + i ^ - 6 j ^ - 7 k ^ ⇒ r → = 1 - 2 + 1 i ^ + - 2 + 4 - 6 j ^ + 1 + 5 - 7 k ^ ⇒ r → = 0 i ^ - 4 j ^ - 1 k ^ Hence, the sum of the vectors is 0 i ^ - 4 j ^ - k ^ .