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Understanding 3D Vectors and Linear Combinations

Learn about 3D unit vectors, linear combinations, and magnitudes in vectors. Discover how to express vectors as linear combinations and solve vector problems in 3D space. Practice with vector displacement examples.

markburns
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Understanding 3D Vectors and Linear Combinations

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  1. Vectors (2) 3D unit vectors Linear Combinations

  2. Magnitude of a 3D Vector (General) t s r z a y |a| = (r2 +s2 + t2) o “the magnitude is the square root of the sum of the squares of the 3 components.” [Pythagoras in 3D] x

  3. Unit Vectors (1) i is the unit vector in the x-direction k j is the unit vector in the y-direction j i k is the unit vector in the z-direction z y x All vectors can be expressed as a linear combination of these 3 vectors

  4. Unit Vectors (2) z y a = i - 2j - 6k x a All 3D Vectors can be expressed in this form

  5. 3D Displacement Vectors z y a = i - 2j - 6k x |a| = (12 +(-2)2 + (-6)2) a |a| = (1+4+ 36) = 41 |a| = 6.4

  6. Linear Combinations 3a + 6b 2a - b

  7. Linear Combination Example B 6i + 4j i - 5j 1 -5 64 7 -1 + = A C The resultant displacement is from A to C A car drives from A to B with displacement = 6i + 4j Then he drives from B to C with displacement =i - 5j (6i + 4j)+(i - 5j) = 7i - j The magnitude of the displacement = (72 + (-1)2) = 50 = 7.1m (1 d.p.)

  8. 3D Vectors Follow all the same rules of 2D Vectors!

  9. 3D Displacement Vectors - adding z y a + b a = i - 2j - 6k x b = 3i + 4j + 11k a b a + b = (i - 2j - 6k) + (3i + 4j + 11k) a + b = (4i + 2j + 5k)

  10. Vector Problem + 13x = 39 Given: a = 3i + 4j andb =i - 3j Find x and y, if xa + yb = 11i + 6j xa + yb = x(3i + 4j) + y(i - 3j) = 3xi + 4xj + yi - 3yj = (3x+y)i + (4x-3y)j = 11i + 6j Therefore: 3x + y = 11 (iparts) x3 9x + 3y = 33 4x - 3y = 6 and: 4x - 3y = 6 (jparts) 3x + y = 11 Substitute x = 3 9 + y = 11 y = 2

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