What vector should be added to the vector a → = 3 i ^ + 4 j ^ - 2 k ^ so that the resultant is a unit vector i ^ ?

Given vector: a → = 3 i ^ + 4 j ^ - 2 k ^ . Let the vector to be added is b → . Given that, a → + b → = i ^ ⇒ b → = i ^ - a → ⇒ b → = i ^ - 3 i ^ + 4 j ^ - 2 k ^ ⇒ b → = - 2 i ^ - 4 j ^ + 2 k ^ Hence, the vector which should be added to the vector a → to get resultant as i ^ is b → = - 2 i ^ - 4 j ^ + 2 k ^ .

Show that the vectors 2 i ^ - 3 j ^ + 4 k ^ and - 4 i ^ + 6 j ^ - 8 k ^ are collinear.

Let the origin be O and points be A and B . Therefore, the position vectors of the points will be, O A → = P . V o f A - P . V o f O Position vector of A is 2 i ^ - 3 j ^ + 4 k ^ . So, O A → = 2 i ^ - 3 j ^ + 4 k ^ - 0 i ^ + 0 j ^ + 0 k ^ ⇒ O A → = 2 i ^ - 3 j ^ + 4 k ^ O B → = P . V o f B - P . V o f O Position vector of B is - 4 i ^ + 6 j ^ - 8 k ^ . So, O B → = - 4 i ^ + 6 j ^ - 8 k ^ - 0 i ^ + 0 j ^ + 0 k ^ ⇒ O B → = - 4 i ^ + 6 j ^ - 8 k ^ ⇒ O B → = - 2 2 i ^ - 3 j ^ + 4 k ^ Here, vectors O A → & O B → are parallel because they are scalar multiples of each other i.e. O B → = - 2 O A → . Their origins are also common. Two vectors are collinear if they have same or parallel supports irrespective of their magnitudes or directions. In this case supports are parallel. Therefore, vectors O A → and O B → are collinear and thus A and B are collinear.

If a → = p i ^ + 3 j ^ and b → = 4 i ^ + p j ^ , then find the values of p so that a → and b → may be parallel.

Let the vectors a → and b → are parallel for some scalar λ . Two vectors are parallel if they are scalar multiples of each other. So, a → = λ b → ⇒ p i ^ + 3 j ^ = λ 4 i ^ + p j ^ ⇒ p i ^ + 3 j ^ = λ 4 i ^ + λ p j ^ On, comparing the coefficients of i ^ & j ^ we get, p = 4 λ , 3 = λ p Here, λ = 3 p So, p = 4 λ ⇒ p = 4 3 p ⇒ p 2 = 12 ⇒ p = ± 12 Hence, the vectors a → and b → are parallel for p = ± 12 .

Write the direction ratios of the vector i ^ + j ^ - 2 k ^ , and hence calculate its direction cosines.

Let r → = i ^ + j ^ - 2 k ^ . Direction ratios are the scalar components of any vector in the x , y & z direction. Here, scalar components are 1 , 1 , - 2 . Therefore, direction ratios are 1 , 1 , - 2 . Direction cosines are the ratio of direction ratio in particular direction and magnitude of that vector. So, Magnitude of r → is, r → = 1 2 + 1 2 + - 2 2 ⇒ r → = 1 + 1 + 4 ⇒ r → = 6 Let the direction cosines are l , m , n . So, l = 1 6 , m = 1 6 , n = - 2 6 Hence, direction cosines of vector r → is 1 6 , 1 6 , - 2 6 .

Find 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 ⇒ r → = 1 + 4 + 9 ⇒ r → = 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 . Hence, direction cosines of r → is 1 14 , 2 14 , 3 14 .