Vector A points to the north and has length A. Vector B points to the east and has length 2.5A. Find the direction of C= 7.4A+B Express your answer as an angle relative to east.

Question

Vector A points to the north and has length A. Vector B points to the east and has length 2.5A. Find the direction of C= 7.4A+B Express your answer as an angle relative to east.

Answer 1

C=7.4A N + 2.5A East

mag c= A sqrt(7.4^2+2.5^2) figure that out.

Angle= arctan 7.4/2.5 N of E

Answer 2

71 degrees N of E

Answer 3

To find the direction of vector C, we will use trigonometry. Let's break vector C into its horizontal and vertical components.

The horizontal component of vector C (Cx) is equal to the horizontal component of vector B plus the horizontal component of vector C:

Cx = Bx + Cx

The vertical component of vector C (Cy) is equal to the vertical component of vector A plus the vertical component of vector B:

Cy = Ay + By

Given that vector B points to the east, it has no vertical component (By = 0) and its horizontal component (Bx) is equal to its magnitude (2.5A).

Therefore, we have:

Cx = 2.5A + Cx
Cy = Ay + 0

Next, we can use the Pythagorean Theorem to relate the lengths of vector C to its horizontal and vertical components:

C^2 = Cx^2 + Cy^2

Substituting the expressions for Cx and Cy:

C^2 = (2.5A + Cx)^2 + Ay^2

Expanding and simplifying:

C^2 = (6.25A^2 + 5Ax + x^2) + Ay^2

Since the magnitude of vector A is A, we can write:

C^2 = 6.25A^2 + 5Ax + x^2 + Ay^2

Now, let's solve for the direction of vector C. We can use the tangent function:

tan(theta) = Cy / Cx

Substituting Ay for Cy and (2.5A + Cx) for Cx:

tan(theta) = Ay / (2.5A + Cx)

Rearranging the equation:

tan(theta) = Ay / (2.5A + Cx)

tangent(theta) = Ay / (2.5A + x)

Therefore, the direction of vector C, relative to east, is given by:

theta = arctan(Ay / (2.5A + x))

θ = arctan(Ay / (2.5A + x))

Please note that the values of Ay and x depend on the specific values of vector A, which are not provided in the question.

Answer 4

To find the direction of the vector C, we can break it down into its components relative to the east (x) and north (y) directions.

Let's start by assigning a coordinate system. We'll use the positive x-axis as east and the positive y-axis as north.

Given that vector A points north, it has no component along the x-axis (east). Therefore, the x-component of A is 0.

The given vector B points east and has a length of 2.5A. This means the x-component of B is 2.5A, and the y-component of B is 0.

To find the components of the vector C, we need to multiply each component of A and B by their scaling factors.

The x-component of C is given by:
Cx = (scaling factor for A) * (x-component of A) + (scaling factor for B) * (x-component of B) = 7.4A + 1*(2.5A) = 7.4A + 2.5A = 9.9A

The y-component of C is given by:
Cy = (scaling factor for A) * (y-component of A) + (scaling factor for B) * (y-component of B) = 7.4A + 0 = 7.4A

Now we have the x-component (9.9A) and the y-component (7.4A) of vector C.

To find the angle of C relative to the east (x-axis), we can use the inverse tangent (arctan) function:

θ = arctan(Cy/Cx) = arctan((7.4A)/(9.9A))

Simplifying:
θ = arctan(7.4/9.9)

Now you can use a calculator or computer software to find the arctan of 7.4/9.9, which gives you the angle in radians or degrees relative to the east.