Explanation:
A unit vector is defined as a vector whose magnitude is exactly one.
It is used only to indicate direction and not strength.
Therefore it has magnitude one and points along a specific direction.

Explanation:
A vector along the x-axis makes angle 0° with that axis.
Its entire magnitude lies along x, so projection on y-axis is zero.
Hence for a vector along the x-axis the y-component vanishes.

Explanation:
Angle 0° means the force lies exactly along the positive x-axis.
So the whole magnitude is in the x-direction and none in the y-direction.
Therefore components are 10 N along x and 0 N along y.

Explanation:
The x-component is the projection of the vector on the x-axis.
In a right triangle, adjacent side to angle θ is given by magnitude times cosθ.
So the x-component of the vector is F cosθ.

Explanation:
A vector along the y-axis makes angle 90° with the x-axis.
The projection of such a vector on the x-axis is zero because cos90° = 0.
So the x-component of a purely vertical vector is zero.

Explanation:

The x-component is
Fₓ = F·cos θ
= 25·cos 180°
= 25·(–1) = –25 N,
whose magnitude is 25 N.

Explanation:
The y-component of a vector at angle 30° is F sin30°.
Sin30° has value 1 / 2.
Therefore the y-component of the force is F / 2.

Explanation:
For perpendicular components, magnitude of the vector is √(Ax² + Ay²).
Using given components,
√(5² + 12²)
= √(25 + 144)
= √169
= 13.
So the components are consistent with the given vector magnitude.

Explanation:

The x-component is F_x = F·cos θ
= 20·cos 60°
= 20·0.5
= 10 N.

Explanation:
When vectors are written as components along axes, each component behaves like a simple scalar quantity.
Components along the same axis can be added or subtracted directly.
This simplifies calculation of resultant vectors while preserving full vector information.

Explanation:
Components along axes are usually smaller than the original vector in magnitude.
However, when combined vectorially, they give back the same original vector.
So resolution does not change the vector, it just expresses it in another convenient form.

Explanation:
Many problems become simpler when a vector is expressed in terms of simpler parts.
This process of splitting a single vector into parts is called resolution of a vector.
Usually these parts are chosen along convenient directions such as coordinate axes.
So resolution means breaking a vector into components, not adding vectors.

Explanation:

A force entirely along x has no y-component ⇒ 0 N.

Explanation:
The x-component is given by magnitude multiplied by cos of the angle with x-axis.
Here it is 20 cos60°.

=20 (1/2)
=10 units

Explanation:
Two-dimensional motion takes place in a plane with two perpendicular axes.
A vector in this plane can be completely described by its x- and y-components.
So it has two independent rectangular components.

Explanation:

Explanation:
When a vector is resolved, it is expressed as the sum of simpler vectors along chosen directions.
These simpler vectors are called components of the original vector.
If the resolution is correct, their vector sum reproduces exactly the original vector in both magnitude and direction.

Explanation:
Unit vectors i and j point along x- and y-axes respectively.
Ax and Ay are scalar multipliers that give the length of A along each axis.
Hence Ax and Ay are the scalar components of A along x and y directions.

Explanation:
At 45°, both cosine and sine have the same value 1 / √2.
So x-component is
F cos45°
= F / √2.
Similarly y-component is
F sin45°
= F / √2.
Thus both components have equal magnitude F / √2.

Explanation:
Magnitude of a vector with components Ax and Ay is √(Ax² + Ay²).
Here magnitude is
√((-4)² + (-3)²).
That equals √(16 + 9)
= √25
= 5 units.

Explanation:

A force along the x-axis makes
θ=0° with the x-axis, so
F_y = F·sin θ
= 10·sin 0°
= 10·0 = 0 N.

Explanation:

Fₓ = F·cos θ
= 8·cos 45°
= 8·(√2/2)
= 4√2 N.

Explanation:
Angle 90° means the force lies along the positive y-axis.
So there is no horizontal component and full magnitude in vertical direction.
Thus components are 0 N along x and 10 N along y.

Explanation:
A vector can be resolved along any two non-collinear directions.
Non-collinear means the two directions are not along the same straight line.
Perpendicular directions are a convenient special case, but not strictly necessary.

Explanation:
Zero y-component means the vector lies along the x-axis.
Along positive x-axis the angle is 0°, and along negative x-axis it is 180°.
So for zero y-component, the angle with positive x-axis is either 0° or 180°.

Explanation:
Weight acts along the vertical direction.
Horizontal direction is perpendicular to the direction of weight.
Projection of a vector on a perpendicular direction is zero.
So horizontal component of weight is zero.

Explanation:
For a vector at angle 30°, x-component is F cos30°.
cos30° is √3 / 2.
So the x-component of the force is F √3 / 2.

Explanation:
A negative x-component points towards the left.
A positive y-component points upwards.
Together they place the vector in the second quadrant of the x–y plane.

Explanation:
A non-zero Ax with Ay equal to zero means there is only horizontal part.
The vector lies completely along the x-axis.
So it is a purely horizontal vector with no vertical component.

Explanation:
The y-component is obtained by projecting the vector on the y-axis.
In the right triangle formed, the opposite side to angle θ is F sinθ.
Therefore the y-component of the vector is F sinθ.

Explanation:
In two-dimensional motion we often use x- and y-axes at right angles to each other.
Expressing a vector along these perpendicular axes gives two independent components.
These perpendicular components can be handled with simple algebra and Pythagoras.

Explanation:

Fₓ = F·cos θ
= 15·cos 90°
= 15·0 = 0 N.

Explanation:
Components Ax and Ay act like the legs of a right-angled triangle.
The original vector is represented by the hypotenuse of this triangle.
By Pythagoras, magnitude of the vector is
√(Ax² + Ay²).

Explanation:
Equal positive components mean Ax = Ay and both are greater than zero.
Then
tanθ = Ay / Ax = 1.
So θ = tan⁻¹(1) = 45°,
measured from the x-axis towards the y-axis.

Explanation:
Horizontal component is given by F cosθ when θ is measured from the horizontal.
Vertical component is F sinθ with the same angle definition.
So resolution into horizontal and vertical components uses both sine and cosine.

Explanation:
In the right triangle for components, tanθ equals opposite over adjacent.
Here opposite side is Ay and adjacent side is Ax.
So
tanθ = Ay / Ax
and θ = tan⁻¹(Ay / Ax)
when Ax is positive.

Explanation:
Component Ax = 0 means there is no horizontal part of the vector.
Ay < 0 means the vertical component points downward. So the vector lies along the negative y-axis.

Explanation:

Fᵧ = F·sin θ
= 12·sin 30°
= 12·0.5
= 6 N.