3f10bd33-befb-47f2-81cb-0fecb156ee75

3f10bd33-befb-47f2-81cb-0fecb156ee75 - SUPPLEMENTARY...

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SUPPLEMENTARY PROBLEMS ON SCALAR & CROSS PRODUCTS 1) Prove that ab ba →→ + is orthogonal (perpendicular) to . 2) Let ai j k →→ → → =+ + 23 , bij k =− + + 2 and ci j =+ 3 . Find λ such that abc + . 3) Find ab × if a = 10 , b = 2 and = 12 . 4) Let aijk →→→→ =++ and bi j k . Find the unit vector perpendicular to + and . 5) Let aij k =+− 45 , and j k 3 . Find vector d such that da , db and cd = 21. 6) Let a = 3 , b = 4 and c = 5 such that each is perpendicular to the sum of the other two (i.e. a , b and c are mutually perpendicular to one another). Show that →→→ ++ = 52 . Hint : Try the substitution uab and use the definition of the scalar product. 7) If →→→ → 0 and a = 3 , b = 5 and c = 7 , show that the angle between a and b is 60 o . Hint : 0 implies that the polygon formed is a triangle. 8) If θ is the angle between a and b , show that tan = × . 9) Prove that 22 2 0 . 10) If ab ac = , ×=× and a 0 , then show bc = . Hint : To show that two vectors are equivalent, you need to prove that their magnitudes are equivalent as well as their directions are equivalent. 11) If a and b are unit vectors and is the angle between them, show that sin 2 1 2 =− . Hint : You need to use the double angle formula for cosine. 12) Let one side of a triangle lie on the diameter of a semi–circle with radius r . Prove that the angle opposite the side that lies along the diameter is a right angle (i.e. ∠= APB 90 o ) . Hint : Use the diagram to the right to find vector equations dealing with PA ⎯→ and PB . Then use the scalar product. O A P B
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SOLUTIONS : 1) Let ua b b a →→ =+ and va b b a =− . Thus, u v ab ba ab ab ab ba ba ab ba ba a b ba ab b a → → →→ →→ →→ → → •• ••••
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3f10bd33-befb-47f2-81cb-0fecb156ee75 - SUPPLEMENTARY...

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