If a → = i ^ - j ^ and b → = - j ^ + k ^ , find the projection of a → on b →

We have a → = i ^ - j ^ and b → = - j ^ + k ^ We know that, the projection of a → on b → is a → · b → b → = i ^ - j ^ · - j ^ + k ^ - j ^ + k ^ = 0 + 1 + 0 1 + 1 = 1 2

If a → = i ^ - j ^ and b → = - j ^ + k ^ , find the projection of a → on b →

We have a → = i ^ - j ^ and b → = - j ^ + k ^ We know that, the projection of a → on b → is a → · b → b → = i ^ - j ^ · - j ^ + k ^ - j ^ + k ^ = 0 + 1 + 0 1 + 1 = 1 2

Write the projections of r → = 3 i ^ - 4 j ^ + 12 k ^ on the coordinate axes.

We have r → = 3 i ^ - 4 j ^ + 12 k ^ Projection of r → on x -axis = r → · i ^ i ^ = 3 1 = 3 Projection of r → on y -axis = r → · j ^ j ^ = - 4 1 = - 4 Projection of r → on z -axis = r → · k ^ k ^ = 12 1 = 12

Write the component of b → along a →

Component of a given vector b → along a → is given by the length of b → on a → . Let θ be the angle between both the vectors. So the length of b → on a → is given as: b → cos θ We know that, The formula to find the angle between two vectors a → and b → is given by cos θ = a → · b → a → b → Therefore, b → cos θ = b → × a → · b → a → b → = a → · b → a → .

Find the value of θ ∈ ( 0 , π / 2 ) for which vectors a → = sin θ i ^ + cos θ j ^ and b → = i ^ - 3 j ^ + 2 k ^ are perpendicular.

We have a → = sin θ i ^ + cos θ j ^ and b → = i ^ - 3 j ^ + 2 k ^ It is given that the vectors are perpendicular ⇒ a → · b → = 0 ⇒ sin θ 1 + cos θ - 3 + 0 2 = 0 ⇒ sin θ - 3 cos θ = 0 ⇒ sin θ = 3 cos θ ⇒ tan θ = 3 ⇒ θ = π 3 .

Write a vector satisfying a → · i ^ = a → · ( i ^ + j ^ ) = a → · ( i ^ + j ^ + k ^ ) = 1

Let a → = a 1 i ^ + a 2 j ^ + a 3 k ^ a → · i ^ = a 1 a → · ( i ^ + j ^ ) = a 1 + a 2 a → · ( i ^ + j ^ + k ^ ) = a 1 + a 2 + a 3 Given that a → · i ^ = a → · ( i ^ + j ^ ) = a → · ( i ^ + j ^ + k ^ ) = 1 ⇒ a 1 = a 1 + a 2 = a 1 + a 2 + a 3 = 1 ⇒ a 1 = 1 ; 1 + a 2 = 1 ; 1 + a 2 + a 3 = 1 ⇒ a 1 = 1 ; a 2 = 0 ; a 3 = 0 So, a → = a 1 i ^ + a 2 j ^ + a 3 k ^ = i ^ .