Write a vector in the direction of vector 5 i ^ - j ^ + 2 k ^ which has magnitude of 8 units .

Given vector: r → = 5 i ^ - j ^ + 2 k ^ . To find: A vector of 8 units in the direction of the vector r → . We know that, unit vector is, r ^ = r → r → . So, unit vector will be, r ^ = 5 i ^ - j ^ + 2 k ^ 5 2 + - 1 2 + 2 2 ⇒ r ^ = 1 30 5 i ^ - j ^ + 2 k ^ So, vector of 8 units is 8 r ^ . = 8 30 5 i ^ - j ^ + 2 k ^ Which is the required vector of 8 units in the direction of r → .

Write the direction cosines of the vector i ^ + 2 j ^ + 3 k ^ .

Let r → = i ^ + 2 j ^ + 3 k ^ . Direction cosines are the ratio of the direction ratio in a particular direction and magnitude of the vector. Here, direction ratios of r → are 1 , 2 , 3 . Magnitude of r → is, r → = 1 2 + 2 2 + 3 2 = 14 Let the direction cosines are l , m , n in x , y , z directions respectively. So, l = 1 14 , m = 2 14 , n = 3 14 . Which are the required direction cosines of the vector.

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

Given, a → = 2 i ^ - 3 j ^ + 6 k ^ . ∴ Unit vector in the direction of a → = a ^ = a → | a → | = 2 i ^ - 3 j ^ + 6 k ^ 2 2 + - 3 2 + 6 2 = 2 i ^ - 3 j ^ + 6 k ^ 49 = 2 i ^ - 3 j ^ + 6 k ^ 7 Which is the required unit vector in direction of given vector

Write the direction cosines of the vectors - 2 i ^ + j ^ - 5 k ^ .

Let r → = - 2 i ^ + j ^ - 5 k ^ . Direction cosines are the ratio of the direction ratio in a particular direction and magnitude of the vector. Here, direction ratios of r → are - 2 , 1 , - 5 . Magnitude of r → is, r → = - 2 2 + 1 2 + - 5 2 = 30 Let the direction cosines are l , m , n in x , y , z directions respectively. So, l = - 2 30 , m = 1 30 , n = - 5 30 . Which are the required direction cosines of the vector.

Find the sum of the following vectors: a → = i ^ - 2 j ^ , b → = 2 i ^ - 3 j ^ , c → = 2 i ^ + 3 k ^ .

Given, a → = i ^ - 2 j ^ , b → = 2 i ^ - 3 j ^ , c → = 2 i ^ + 3 k ^ . So, Sum of the three vectors = a → + b → + c → = i ^ - 2 j ^ + 2 i ^ - 3 j ^ + 2 i ^ + 3 k ^ = 5 i ^ - 5 j ^ + 3 k ^

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

We have, a → = 3 i ^ - 2 j ^ + 6 k ^ ∴ Unit vector in the direction of a → = a ^ = a → | a → | = 3 i ^ - 2 j ^ + 6 k ^ 3 i ^ - 2 j ^ + 6 k ^ = 3 i ^ - 2 j ^ + 6 k ^ 3 2 + - 2 2 + 6 2 = 3 i ^ - 2 j ^ + 6 k ^ 49 = 1 7 ( 3 i ^ - 2 j ^ + 6 k ^ ) Which is the required unit vector in direction of given vector