Question

Often a vector is specified by a magnitude and a direction; for example, a rope with tension T⃗ exerts a force of magnitude T=20N in a direction θ=35∘ north of east. This is a good way to think of vectors; however, to calculate results with vectors, it is best to select a coordinate system and manipulate the components of the vectors in that coordinate system.Find the components of the vector A⃗ with length a = 1.00 and angle α=20.0 ∘ with respect to the x axis as shown in (Figure 1) .

Answer 1

Well first of all, I take issue with the opening statement of your question.
It's vital to learn and understand that a vector is ALWAYS specified by a magnitude and a direction.  "OFTEN" is dangerously misleading, especially
at the high school level.

Next, the question technically dies at the end, where it says "... as shown
in Figure 1", since there's no Figure 1 to be found.   But since I've already
gone this far, I think I can work with it.

If 20° is the angle between the vector A and the x-axis, then . . .

-- the x-component of the vector is  (magnitude of A) cos(20)

-- the y-component of the vector is (magnitude of A) sin(20).

The magnitude of A is 1.00, so the components are

-- x-component = cos(20)  =  0.9397  (rounded)

-- y-component = sin(20)  =  0.342  (rounded)

Answer 2

The components of the vector A :

• Ax = 0.940
• Ay = 0.342

Further explanation

In the coordinates of the two-dimensional plane (xy) there is a position vector.

Two dimensions because the coordinates are only x and y, three dimensions if the coordinates are x, y, and z.

If there is a point P then

You can write P as:

P = xi + yi

where i denotes the unit vector in the x and y directions

While the size of the vector:

$\displaystyle P = \sqrt {x ^ 2 + y ^ 2}$

and its direction

$\displaystyle \theta = arc \: tan \: \frac {y} {x}$

Each vector can be described in two perpendicular vectors

The first vector is a vector component on the x-axis (Fx) and the second vector is a vector component on the y-axis (Fy)

Using the principle rules of cosines and sines in triangles then:

Fx = F cos θ

Fy = F sin θ

The magnitude of a vector is always positive, while the angle formed (θ) can be both positive and negative depending on the quadrant (direction) of the vector in the Cartesian coordinate system

The components of the vector A with length a = 1.00 and angle α = 20° with respect to the x axis, so

The magnitude of vector A = 1

θ = 20 °

Then the components Ax and Ay

Ax = A cos θ

Ax = 1. cos 20 °

Ax = 0.940

Ay = A sin θ

Ay = 1 sin 20 °

Ay = 0.342

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