Question

Let →=⟨1,−1,−3,1⟩ a → = ⟨ 1 , − 1 , − 3 , 1 ⟩ and →=⟨−1,2,1,4⟩∈ℝ4 b → = ⟨ − 1 , 2 , 1 , 4 ⟩ ∈ r 4 with weighted euclidean inner product corresponding to weights 1=1,2=2,3=3,4=1 w 1 = 1 , w 2 = 2 , w 3 = 3 , w 4 = 1 . Find each of the following values, showing your work. a.) ⟨→,→⟩ ⟨ a → , b → ⟩ b.) ‖‖‖→ →‖‖‖ ‖ a → b → ‖ c.) (→,→) d ( a → , b → ) d.) proj→→

Answer 1

a) The inner product of the given vectors is 12. b) The norm of the given vector is √12. c) The inner product of a vector with itself is 12. d) The projection of the given vector onto itself is (1,-1,-3,1).

Step-by-step explanation:

a) We can find the inner product of two vectors by taking the sum of the products of their corresponding components. For example, to find ⟨→,→⟩ , we multiply the corresponding components of the vectors and add them together: 1*1 + (-1)*(-1) + (-3)*(-3) + 1*1 = 12.

b) To find the norm of a vector, we take the square root of the sum of the squares of its components. For example, to find the norm of →→→→ , we calculate the square root of 1^2 + (-1)^2 + (-3)^2 + 1^2 = √12.

c) The inner product of a vector with itself is equal to the square of its norm. So, (→,→) = √12^2 = 12.

d) The projection of one vector onto another can be found using the formula: proj→→ = (→,→)/(→,→) * →→→ = 12/12 * (1,-1,-3,1) = (1,-1,-3,1).