1. 2. 3. 4. 5. Let u (2, 3, -4) and v - = (2,5, -1)...

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1. 2. 3. 4. 5. Let u (2, 3, -4) and v - = (2,5, -1) be two vectors of size 3. (a) Determine the cosine of the angle between u and v. (b) Find the unit vector w₁ in the direction of u. (c) Find the unit vector w₂ in the direction of v. Let u, V. and w be three vectors in R³. Suppose that u and v are orthogonal to w. Using the properties of inner product: (a) Show that u + v is orthogonal to w. (b) Show that for any scalar a, au is also orthogonal to w. Let W be the line with equation y = 3 - 2x. Is W a subspace of R2? Justify your answer. Let W = a a + c C : a, b, c ER be a subset of R4. (a) Show that W is a subspace of R4. (b) Find a set S of 4-vectors that spans W. Let P be the plane in R³, with equation 2xy +z = 0. (a) Show that P is a subspace of R³. (b) Find a set S of 3-vectors that spans W. Hints: For question 3 (similar to question 5), any vector in W is of the form (x, y) where y = 32x, i.c., any vector in W is of the form (x, 3 — 2x). - 1. 2. 3. 4. 5. Let u (2, 3, -4) and v - = (2,5, -1) be two vectors of size 3. (a) Determine the cosine of the angle between u and v. (b) Find the unit vector w₁ in the direction of u. (c) Find the unit vector w₂ in the direction of v. Let u, V. and w be three vectors in R³. Suppose that u and v are orthogonal to w. Using the properties of inner product: (a) Show that u + v is orthogonal to w. (b) Show that for any scalar a, au is also orthogonal to w. Let W be the line with equation y = 3 - 2x. Is W a subspace of R2? Justify your answer. Let W = a a + c C : a, b, c ER be a subset of R4. (a) Show that W is a subspace of R4. (b) Find a set S of 4-vectors that spans W. Let P be the plane in R³, with equation 2xy +z = 0. (a) Show that P is a subspace of R³. (b) Find a set S of 3-vectors that spans W. Hints: For question 3 (similar to question 5), any vector in W is of the form (x, y) where y = 32x, i.c., any vector in W is of the form (x, 3 — 2x). -