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# A(2, 3), B(−1, 5), C(−1, 1) and D(−7, 5) are four points in the Cartesian plane, Check if, CD¯ is parallel to AB¯ - Mathematics and Statistics

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A(2, 3), B(−1, 5), C(−1, 1) and D(−7, 5) are four points in the Cartesian plane, Check if, bar("CD") is parallel to bar("AB")

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#### Solution

Let bar("a"), bar("b"), bar("c"), bar("d") be the position vectors of points A, B, C, D respectively.

∴ bar("a") = 2hat"i" + 3hat"j", bar("b") = -hat"i" + 5hat"j", bar("c") = -hat"i" + hat"j", bar("d") = -7hat"i" + 5hat"j"

bar("CD") = bar("d") - bar("c")

= (-7hat"i" + 5hat"j") - (-hat"i" + hat"j")

= -6hat"i" + 4hat"j"

= 2(-3hat"i" + 2hat"j")

bar("AB") = bar("b") - bar("a")

= (-hat"i" + 5hat"j") - (2hat"i" + 3hat"j")

= 3hat"i" + 2hat"j"

As bar("CD") = 2bar("AB"), bar("CD") and bar("AB") are parallel.

Concept: Vector Joining Two Points
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