# Let bar"b" = 4hat"i" + 3hat"j" and bar"c" be two vectors perpendicular to each other in the XY-plane. Find the vector in the same plane having projection 1 and 2 along bar"b" and bar"c" respectively. - Mathematics and Statistics

Sum

Let bar"b" = 4hat"i" + 3hat"j" and bar"c" be two vectors perpendicular to each other in the XY-plane. Find the vector in the same plane having projection 1 and 2 along bar"b" and bar"c" respectively.

#### Solution

bar"b" = 4hat"i" + 3hat"j"

∴ |bar"b"| = sqrt(4^2 + 3^2) = sqrt(16 + 9) = 5

Let bar"c" = "m"hat"i" + "n"hat"j" be perpendicular to bar"b"

Then bar"b".bar"c" = 0

∴ (4hat"i" + 3hat"j").("m"hat"i" + "n"hat"j") = 0

∴ 4m + 3n = 0

∴ n = - "4m"/3

∴ bar"c" = "m"hat"i" - "4m"/3hat"j" = "m"/3(3hat"i" - 4hat"j")

∴ bar"c" = "p"(3hat"i" - 4hat"j")     .....["p" = "m"/3]

∴ |bar"c"| = "p" sqrt(3^2 + (- 4)^2) = "p"sqrt(9 + 16) = 5"p"

Let bar"d" = "x"hat"i" + "y"hat"j" be the vector having projections 1 and 2 along bar"b" and bar"c".

∴ (bar"b".bar"d")/|bar"b"| = 1

∴ ((4hat"i" + 3hat"j").("x"hat"i" + "y"hat"j"))/5 = 1

∴ 4x + 3y = 5          .....(1)

Also, (bar"c".bar"d")/|bar"c"| = 2

∴ ((3"p"hat"i" - 4"p"hat"j").("x"hat"i" + "y"hat"j"))/"5p" = 2

∴ 3px - 4py = 10p

∴ 3x - 4y = 10

From (1), 3y = 5 - 4x

∴ y = (5 - 4"x")/3

Substituting for y in (2), we get

3"x" - 4((5 - "4x")/3) = 10

∴ 9x - 20 + 16x = 30

∴ 25x = 50

∴ x = 2

y = (5 - 4"x")/3 = (5 - 4(2))/3 = - 1

∴ bar"d" = 2hat"i" - hat"j"

Hence, the required vector is 2hat"i" - hat"j".

Concept: Vectors and Their Types
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