A vector 3i+4j rotates about its tail an angle 37¡ã anticlockwise direction ...the new vector is

Physics purposes

To find the new vector after rotating a vector in the anticlockwise direction, you can use a rotation matrix. The rotation matrix for a given angle of rotation in the plane is:

[R] = | cosθ -sinθ |
| sinθ cosθ |

In this case, the angle of rotation is 37 degrees (37°).

Let's start by converting the angle to radians because trigonometric functions in most programming languages work with radians rather than degrees.

37 degrees (37°) is equal to (37 * π) / 180 radians.

Now, we can substitute the values into the rotation matrix.

[R] = | cos((37 * π) / 180) -sin((37 * π) / 180) |
| sin((37 * π) / 180) cos((37 * π) / 180) |

Multiplying the rotation matrix with the given vector (3i + 4j) will give the new rotated vector.

Let's calculate the new vector:

[N] = [R] * [V]

where [N] is the new vector and [V] is the given vector.

[N] = | cos((37 * π) / 180) -sin((37 * π) / 180) | * |3|
| sin((37 * π) / 180) cos((37 * π) / 180) | |4|

Now, let's calculate each component of the new vector:

[N] = | cos((37 * π) / 180) * 3 - sin((37 * π) / 180) * 4 |
| sin((37 * π) / 180) * 3 + cos((37 * π) / 180) * 4 |

Simplifying the equations, we get:

[N] = [(-0.515) * 3 - (-0.857) * 4]
| (0.857) * 3 + (-0.515) * 4 |

[N] = [-2.939 + 3.428]
| 2.571 + (-2.06) |

[N] = [0.489]
| 0.511 |

Therefore, the new vector after rotating 3i + 4j anticlockwise by 37 degrees is approximately 0.489i + 0.511j.