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 .

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 ^ .

Find a unit vector in the direction of the vector a → = 2 i ^ - 3 j ^ .

Given vector a → = 2 i ^ - 3 j ^ . To find: a unit vector in the direction of the vector a → . The vector whose magnitude is one is called unit vector. Unit vector is denoted by a ^ , where a ^ = a → a → So, magnitude of vector a → is a → = 2 2 + - 3 2 ⇒ a → = 4 + 9 = 13 Unit vector will be a ^ = 2 i ^ - 3 j ^ 13 ⇒ a ^ = 2 13 i ^ - 3 13 j ^ Whhich is the unit vector in the direction of vector a ^ .

Find a unit vector in the direction of the vector a → = 2 i ^ + j ^ + 2 k ^ .

Given vector a → = 2 i ^ + j ^ + 2 k ^ . Unit vector is vector whose magnitude is one. Unit vector is represented as a ^ , where a ^ = a → a → So, magnitude of vector a → is a → = 2 2 + 1 2 + 2 2 ⇒ a → = 4 + 1 + 4 ⇒ a → = 9 = 3 Therefore, a ^ = 2 i ^ + j ^ + 2 k ^ 3 ⇒ a ^ = 2 3 1 ^ + 1 3 j ^ + 2 3 k ^ Which is the unit vector in the direction of vector a → .

Find a unit vector in the direction of the vector a → = 3 i ^ - 2 j ^ + 6 k ^ .

Given vector: a → = 3 i ^ - 2 j ^ + 6 k ^ . A vector whose magnitude is one is known as unit vector. Unit vector is represented as a ^ , where a ^ = a → a → So, magnitude of vector a → is a → = 3 2 + - 2 2 + 6 2 ⇒ a → = 9 + 4 + 36 = 49 ⇒ a → = 7 Therefore, a ^ = 3 i ^ - 2 j ^ + 6 k ^ 7 ⇒ a ^ = 3 7 i ^ - 2 7 j ^ + 6 7 k ^ Hence, unit vector in the direction of vector a → is 3 7 i ^ - 2 7 j ^ + 6 7 k ^ .