Question

You are given a vector in the xy plane that has a magnitude of 82.0 units and a y component of -58.0 units.

Part A

What are the two possibilities for its x component?

Enter your answers using three significant figures separated by a comma.

Part B

Assuming the x component is known to be positive, specify the magnitude of the vector which, if you add it to the original one, would give a resultant vector that is 80.0 units long and points entirely in the −x direction.

Part C

Specify the direction of the vector.

Express your answer using three significant figures.

Answer 1

Part A..

Let, Given vector = R = Rx i + Ry j

here, |R| = 82.0 units

Ry = -58.0

So,

|R|^2 = Rx^2 + Ry^2

Rx^2 = 82.0^2 - (-58.0)^2 = 3360

Rx = sqrt(3360) = 57.97

Rx = 58.0 unit

Rx = -58.0, +58.0

Part-B.

Given, x-component is known to be positive.

then, R = 58.0 i - 58.0 j

Let, anothe vector is,

R' = R'x i + R'y j

given,

R + R' = -80 i + 0 j

So,

(58.0 i - 58.0 j) + (R'x i + R'y j) = -80 i + 0 j

(58.0 + R'x) i + (-58.0 + R'y) j = -80 i + 0 j

equating both side,

58.0 + R'x = -80

R'x = -80.0-58.0 = -138

-58.0 + R'y = 0

R'y = 58.0 j

then,

R' = -138 i + 58.0 j

Magnitude will be:

|R'| = sqrt(138^2 + 58.0^2)

|R'| = 150

Part-C.

direction will be:

= arctan(R'y/R'x)

= arctan(58.0/(-138))

= 22.8 deg

Part A..

Let, Given vector = R = Rx i + Ry j

here, |R| = 82.0 units

Ry = -58.0

So,

|R|^2 = Rx^2 + Ry^2

Rx^2 = 82.0^2 - (-58.0)^2 = 3360

Rx = sqrt(3360) = 57.97

Rx = 58.0 unit

Rx = +58.0, -58.0

Part-B.

Given, x-component is known to be positive.

then, R = 58.0 i - 58.0 j

Let, anothe vector is,

R' = R'x i + R'y j

given,

R + R' = -80 i + 0 j

So,

(58.0 i - 58.0 j) + (R'x i + R'y j) = -80 i + 0 j

(58.0 + R'x) i + (-58.0 + R'y) j = -80 i + 0 j

equating both side,

58.0 + R'x = -80

R'x = -80.0-58.0 = -138

-58.0 + R'y = 0

R'y = 58.0 j

then,

R' = -138 i + 58.0 j

Magnitude will be:

|R'| = sqrt(138^2 + 58.0^2)

|R'| = 150

Part-C.

direction will be:

= arctan(R'y/R'x)

= arctan(58.0/(-138))

= -22.8 deg above -x axis.

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