In kinematic calculations and visual computations in robotics, vector operations are expressed with matrices sometimes. Answer the following questions on inner-product, outer-product, projection and rotation of three dimensional vectors. is the identity matrix. The three-dimensional vectors , , and are column vectors:
, which is a row vector, shows the transpose of .
(1) On the inner product between vectors and , describe i) the value of , ii) a matrix which satisfies , and iii) with vector and its transpose .
(2) On the outer product from to , describe i) expression of , ii) expression of the matrix which satisfies , iii) the matrix which satisfies where between a vector and a matrix means a matrix whose column vectors are three outer products from the vector to each column vector in the matrix respectively.
(3) As Figure 1 shows, a vector is vertically projected to a vector on a plane whose normal vector is a unit vector . If the vector is described as , show that the matrix becomes .
(4) Three rotational matrices , and are rotational matrices which rotate a vector around the X-axis, Y-axis and Z-axis with , and respectively, where the direction of the rotation for plus is clock-wise around the axis from the origin to infinity.
i) Describe expression of the matrix , ii) As Figure 2 shows, the matrix is defined as the rotation matrix around a unit orientation vector with . is described as
Explain what the variables and become and explain why the expression is satisfied.
