Vector has a magnitude of 5.10 units and points due east. Vector points due north. (a) What is the magnitude of , if the vector + points 27.4 ° north of east? (b) Find the magnitude of + .

To solve this problem, we will use vector addition and trigonometry. Let's break down the steps to find the magnitudes of the vectors:

(a) Find the magnitude of -

1. The given vector has a magnitude of 5.10 units and points due east.
2. This means the x-component of is 5.10 units and the y-component is 0 units (no north/south displacement).
3. The vector points due north, which means its x-component is 0 units and the y-component is its magnitude.
4. To find the magnitude of , we need to find the x and y components of the vector sum of and .

We can use the formula: Magnitude of vector = sqrt(x^2 + y^2)

5. To find the x-component of the vector sum, we add the x-components of and :

x = 5.10 + 0 = 5.10

6. To find the y-component of the vector sum, we add the y-components of and :

y = 0 + magnitude of = 0 + 5.10 = 5.10

7. Now we can find the magnitude of the vector sum:

Magnitude of = sqrt(5.10^2 + 5.10^2) = sqrt(2*5.10^2) = sqrt(2)*(5.10) ≈ 7.225 units

Therefore, the magnitude of is approximately 7.225 units.

(b) Find the magnitude of +

8. We already have the x and y components of from step 5: x = 5.10 and y = 5.10.
9. To find the magnitude of , we again use the formula: Magnitude of vector = sqrt(x^2 + y^2).
10. Calculate:

Magnitude of = sqrt(5.10^2 + 5.10^2) = sqrt(2*5.10^2) = sqrt(2)*(5.10) ≈ 7.225 units

Therefore, the magnitude of + is approximately 7.225 units.

To solve part (a), we first need to find the components of the given vectors in the east (x) and north (y) direction. Let's call the vector pointing due east as Vector A, and the vector pointing due north as Vector B.

Vector A: Magnitude = 5.10 units, Direction = due east (0°)

The x-component of Vector A will be its magnitude multiplied by the cosine of its direction:
Ax = 5.10 * cos(0°) = 5.10 * 1 = 5.10

The y-component of Vector A will be its magnitude multiplied by the sine of its direction:
Ay = 5.10 * sin(0°) = 5.10 * 0 = 0

Vector B: Magnitude = unknown, Direction = due north (90°)

The x-component of Vector B will be its magnitude multiplied by the cosine of its direction:
Bx = unknown * cos(90°) = unknown * 0 = 0

The y-component of Vector B will be its magnitude multiplied by the sine of its direction:
By = unknown * sin(90°) = unknown * 1 = unknown

Now, Vector C = Vector A + Vector B, and it points 27.4° north of east.

The x-component of Vector C will be the sum of the x-components of Vector A and Vector B:
Cx = Ax + Bx = 5.10 + 0 = 5.10

The y-component of Vector C will be the sum of the y-components of Vector A and Vector B:
Cy = Ay + By = 0 + unknown = unknown

To find the magnitude of Vector C, we can use the Pythagorean theorem:
|C| = √(Cx^2 + Cy^2) = √(5.10^2 + unknown^2)

Now, let's move to part (b) where we need to find the magnitude of Vector A + Vector B.

The x-component of Vector A + Vector B will be the sum of the x-components of Vector A and Vector B:
(A + B)x = Ax + Bx = 5.10 + 0 = 5.10

The y-component of Vector A + Vector B will be the sum of the y-components of Vector A and Vector B:
(A + B)y = Ay + By = 0 + unknown = unknown

To find the magnitude of Vector A + Vector B, we can use the Pythagorean theorem:
|(A + B)| = √((A + B)x^2 + (A + B)y^2) = √(5.10^2 + unknown^2)

To get the values of unknown in both parts (a) and (b), we need additional information or equations that relate the magnitudes and directions of the vectors involved.