Anonymous
Anonymous asked in Science & MathematicsMathematics · 2 years ago

TRIGONOMETRY Let u = 8i + 7j and v = 7i − j.?

(a) Find the angle between u and v. (Round your answer to one decimal place.)

(b) Find the component of u along v.

(c) Find projvu.

2 Answers

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  • 2 years ago
    Favorite Answer

    u = 8i + 7j → tan(u) = 7/8

    v = 7i - j → tan(v) = - 1/7

    tan(a - b) = sin(a - b) / cos(a - b) → recall the identity: sin(a - b) = sin(a).cos(b) - cos(a).sin(b)

    tan(a - b) = [sin(a).cos(b) - cos(a).sin(b)] / cos(a - b) → recall the identity: cos(a - b) = cos(a).cos(b) + sin(a).sin(b)

    tan(a - b) = [sin(a).cos(b) - cos(a).sin(b)] / [cos(a).cos(b) + sin(a).sin(b)] → you divide by cos(b)

    tan(a - b) = [{sin(a).cos(b)/cos(b)} - {cos(a).sin(b)/cos(b)}] / [{cos(a).cos(b)/cos(b)} + {sin(a).sin(b)/cos(b)}]

    tan(a - b) = [sin(a) - cos(a).tan(b)] / [cos(a) + sin(a).tan(b)] → you divide by cos(a)

    tan(a - b) = [{sin(a)/cos(a)} - {cos(a).tan(b)/cos(a)}] / [{cos(a)/cos(a)} + {sin(a).tan(b)/cos(a)}]

    tan(a - b) = [tan(a) - tan(b)] / [1 + tan(a).tan(b)] → adapt this result to your case

    tan(u - v) = [tan(u) - tan(v)] / [1 + tan(u).tan(v)]

    tan(u - v) = [(7/8) - (- 1/7)] / [1 + (7/8).(- 1/7)]

    tan(u - v) = [(7/8) + (1/7)] / [1 - (1/8)]

    tan(u - v) = [(49/56) + (8/56)] / [(8/8) - (1/8)]

    tan(u - v) = (57/56) / (7/8)

    tan(u - v) = (57/56) * (8/7)

    tan(u - v) = (57 * 8) / (56 * 7)

    tan(u - v) = (57 * 8) / (8 * 7 * 7)

    tan(u - v) = 57/49

    tan(u - v) ≈ 1.163265

    u - v ≈ 49.316 °

    u.v = IIuII * IIvII * cos(u ; v) → recall: u = 8i + 7j → IIuII = √[(8)² + (7)²] = √(64 + 49) = √113

    u.v = √113 * IIvII * cos(u ; v) → recall: v = 7i - j → IIuII = √[(7)² + (- 1)²] = √(49 + 1) = √50 = 5√2

    u.v = √113 * 5√2 * cos(u ; v)

    u.v = 5√226 * cos(u ; v) ← memorize this result

    tan(u - v) = 57/49

    tan(u - v) = sin(u - v)/cos(u - v)

    sin(u - v) = cos(u - v).tan(u - v)

    sin²(u - v) = cos²(u - v).tan²(u - v)

    Recall the famous formula: cos²(x) + sin²(x) = 1

    cos²(u - v) + sin²(u - v) = 1 → recall the previous result

    cos²(u - v) + cos²(u - v).tan²(u - v) = 1

    cos²(u - v).[1 + tan²(u - v)] = 1

    cos²(u - v) = 1/[1 + tan²(u - v)] → recall: tan(u - v) = 57/49

    cos²(u - v) = 1/[1 + (57/49)²]

    cos²(u - v) = 1/[(49²/49²) + (57²/49²)]

    cos²(u - v) = 1/[(49² + 57²)/49²]

    cos²(u - v) = 49²/(49² + 57²)

    cos²(u - v) = 49²/5650

    cos(u - v) = 49/√5650

    cos(u - v) = 49/(5√226)

    Restart with the memorized result:

    u.v = 5√226 * cos(u ; v) → we've just seen that: cos(u - v) = 49/(5√226)

    u.v = 5√226 * [49/(5√226)]

    u.v = 49

  • alex
    Lv 7
    2 years ago

    Rule:

    u = <a,b> , v=<c,d>

    u.v = |u||v|cos(u , v) = ac+bd

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