If the vectors u = (1,2) and v= (2,-4) are orthogonal with respect to the weighted Euclidean inner product (u, v) = wiuiVi + w 2 u 2 v 2 , what must be true of the weights Wi and wp
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Orthogonality implies that the inner product (u, v) = 0. Show more…
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If the vectors $\mathbf{u}=(1,2)$ and $\mathbf{v}=(2,-4)$ are orthogonal with respect to the weighted Euclidean inner product $\langle\mathbf{u}, \mathbf{v}\rangle=w_{1} u_{1} v_{1}+w_{2} u_{2} v_{2},$ what must be true of the weights $w_{1}$ and $w_{2} ?$
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Angle and Orthogonality in Inner Product Spaces
Determine whether the following vectors in R³ are orthogonal with respect to the Euclidean inner product: (a) u = [-1, 3, 2]ᵀ, v = [4, 2, 1]ᵀ (b) u = [-2, -2, -2]ᵀ, v = [1, 1, 1]ᵀ (c) u = [a, b, c]ᵀ, v = [-c, 0, a]ᵀ Show that the following vectors in P₂ are orthogonal with respect to the standard inner product: p = -1 - x + 2x², q = 2x + x² If the vectors u = [1, 2]ᵀ and v = [2, -4]ᵀ are orthogonal with respect to the weighted inner product ⟨u, v⟩ = w₁u₁v₁ + w₂u₂v₂, what must be true about the weights w₁, w₂?
Sri K.
Show that the vectors are not orthogonal with respect to the Euclidean inner product on $R^{2}$, and then find a value of $k$ for which the vectors are orthogonal with respect to the weighted Euclidean inner product $(\mathbf{u}, \mathbf{v})=2 u_{1} v_{1}+k u_{2} v_{2}$. $$\mathbf{u}=(1,3), \quad \mathbf{v}=(2,-1)$$
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