Question

Let u=⟨8,0,−7⟩.

(a) Find the vector and scalar projections of u onto i.
(b) Find the vector and scalar projections of u onto j.
(c) Find the vector and scalar projections of u onto k

Answer 1

The question involves finding the vector and scalar projections of the vector u onto the standard basis unit vectors i, j, and k. The results are: for i, vector projection ⟨8,0,0⟩ and scalar projection 8; for j, both vector and scalar projections are 0; for k, vector projection ⟨0,0,-7⟩ and scalar projection -7.

Step-by-step explanation:

The question asks for the vector and scalar projections of the vector u onto the unit vectors i, j, and k. The vector u is given as u = ⟨8,0,−7⟩. The vector projection of u onto any unit vector is the dot product of u with that unit vector times the unit vector itself. The scalar projection is simply the dot product of u with the unit vector.

Vector and Scalar Projections onto i, j, and k

• i Vector Projection: ⟨8,0,0⟩, Scalar Projection: 8
• j Vector Projection: ⟨0,0,0⟩, Scalar Projection: 0
• k Vector Projection: ⟨0,0,−7⟩, Scalar Projection: −7

The vector projections are parallel to the respective unit vectors and the scalar projections are the lengths of these vector projections in the direction of the unit vectors.