Education Journalist | Study Abroad Lead
Any vector directed in two dimensions can be understood to have 2 different directions of effect. This indicates that it may be divided into two sections. Components are the parts of a two-dimensional vector. The components of a vector are used to represent the vector's effect in a certain direction. The two vector components can be used to replace the single two-dimensional vector. In a two-dimensional coordinate system, the x-component and the y-component are commonly regarded to represent the components of a vector. It can be expressed as,
V = (vx,vy)
| Table of Content |
Key Terms: Scalar, coordinate system, 3-D Vector, 2-D Vector, Pythagoras Theorem, Coordinate Geometry, vector, vector component
Also read: Introduction to Three-dimensional Geometry
Components Of Vector
[Click Here for Sample Questions]
The split of a vector is determined by its components. We can compute the components of a vector by splitting the vector concerning each of the axes. A vector's various components can subsequently be concatenated to form the whole vector representation. In general, vectors are represented in a two-dimensional coordinate plane with an x-axis and y-axis, or a three-dimensional space with the x-axis, y-axis, and z-axis. Vectors are directional and magnitude-based mathematical representations.
In a two-dimensional coordinate system, the vector's direction is determined by the angle it makes with the positive x-axis. Let V be the vector, and be the angle formed by V with the positive x-axis. The components of this vector are labeled Rx and Ry on the x and y axes, respectively. The following equations can be used to calculate these components.
Vx = R.Cosθ, and Ry = R.Sinθ
|V| = √ Rx2 + Ry2
The vectors are also represented as A= ai+bj+ck
in three-dimensional space. The unit vectors for the x-axis, y-axis, and z-axis are shown as i,j,k. With relation to each of the axes, these unit vectors aid in distinguishing the components of the vectors. The components of vector A are a, b, and c, respectively, about the x-axis, y-axis, and z-axis.
Also Read:
| Related Links | ||
|---|---|---|
| Sphere Formula | Quadrilateral Formula | Trapezoid Formula |
| Calculus Formula | Definite Integral Formula | Differential Equation |
Components of a vector formula
[Click Here for Sample Questions]
Since we derived the following formula in the previous section:
cosθ = vx/V
sinθ = vy/V
As a result, the formula for determining the components of any given vector is:
vx=V cos θ
vy=Vsin θ
Read More: Resultant Vector Formula
Components of a Two-Dimensional Vector
[Click Here for Sample Questions]
Consider the case of a two-dimensional vector A, which has a beginning point O and a final point A in the coordinate system. You'll receive two newly formed vectors Ax and Ay if you draw lines from the points O and A that intersect at a point C and establish a 90-degree angle with one another.
Components of a Three-Dimensional Vector
[Click Here for Sample Questions]
If you resolve the provided vector A into its components in the three-dimensional system with the x, y, and z axes, you obtain Ax, Ay, Az, much as the two-dimensional components.
Also Read: Magnitude of A Vector
Things To Remember
- The magnitude and direction of a vector may be split down into two parts.
- You may determine the horizontal and vertical components of a vector using the hypotenuse technique by calculating the angle that the vector creates with the two components.
- Scalar quantities (for example, mass, height, volume, and area) are physical quantities that are represented by a single number, whereas vector quantities (for example, velocity, displacement, and acceleration) are physical values that have two components: direction and magnitude.
- Vector values may be split down into horizontal and vertical axis components.
- A unit vector is defined as a vector with a magnitude of one.
- Because vectors are primarily arrows with a magnitude and direction, any quantity represented by a vector has both magnitude and direction.
- Displacement, acceleration, and velocity are the most frequent physical variables that are expressed as vectors.
- Because acceleration is the rate at which velocity changes for time, it necessitates both direction and magnitude.
Also Read:
Sample Questions
Ques. Find the x and y components of a vector having a magnitude of 12 and make an angle of 45 degrees with the positive x-axis. (3 Marks)
Ans. The given vector is V= 12, and it makes an angle θ = 45º.
The x component of the vector =
Vx = VCosθ = 12.Cos45º = 12.(1/√2) = 6√2.
The y component of the vector =
Vy = VSinθ = 12.Sin45º = 12.(1/√2) = 6√2.
Therefore, the x component and the y components of the vector are both equal to 6√2.
Ques. Find the vector from the components of a vector, having the x-component of 5 units, y-component of 12 units, and z-component of 4 units respectively. (3 Marks)
Ans. Given
X Component of the vector = a = 5
Y Component of th vector = b = 12
Z Component of the vector = c = 4
The required vector is
V= ai + bj + ck
Hence, V= 5i + 12j + 4k
Therefore the required vector is
V= 5i + 12j + 4k
Ques. The magnitude of a given vector F and the direction of its vector is 60along the horizontal. Find its vector components. (3 Marks)
Ans. Fx = FCos60
10 × 1/2 = 5
Fy = Fsin60
10 × √3/2 = 5√3
Hence, the vector F is equal to 5,5√ 3
Ques. A force of 20 N makes an angle of 30 degrees with the x-axis. Find both the x-component and the y component of the given force. (3 Marks)
Ans. Fx= F cos 30 and Fy= F sin 30.
Fx= F cos 30 = 20 x cos 30
= (20)(0.5√3)
= 10√3 Newton
Fy = F sin 30 = 20 x sin 30
= 20 x 0.5
= 10 Newton
Ques. Calculate the magnitude of vector v = (3,8). (3 Marks)
Ans. |v| = √((vx )2+( vy)2)
Where vx = 3 , vy =8
Putting into the formula give
|v| = √((3)2+(8)2)
|v| = 8.544
Ques. A force of 12N is acting on a boat at an angle of 51o with the horizontal. Resolve into its components and prove by using the formula that the magnitude of the force is 12N. (4 Marks)
Ans. we know that,
Fx= F.cosθ
Fx= 12.cos51
Fx= 8.91N
Fy = F.sinθ
Fy = 12.sin51
Fy = 8.04N
Prove that the magnitude of the force in the question is 12N using the magnitude formula.
Using formula,
|F| = √ ((Fx )2+
( Fy)2)
|F| = √ ((8.91 )2+
( 8.04)2)
|F|=12.00N
As a result, the magnitude of the force may be determined using the formula.
Ques. Find out the magnitude and direction of a vector OP= (-4,6). (3 Marks)
Ans. The magnitude of the vector is defined as,
|OP| = √ ((-4)2 +(6)2)
|OP| = √ (16+36)
|OP| = 7.21
The direction of the given vector is,
φ = tan-1 (6/4)
φ = 56.3º
Because the x-component is negative and the y-component is positive, it falls into the second quadrant and is written as, according to the convention described above.
θ = 180º – φ
θ = 180º – 56.3º
θ = 123.7º
Ques. A laser beam is aimed 15.95° above the horizontal at a mirror 11,648 m away. It glances off the mirror and continues for an additional 8570. m at 11.44° above the horizon until it hits its target. What is the resultant displacement of the beam to the target? (5 Marks)
Ans. It's necessary to deal with vectors that aren't at good angles. Break them down into their constituent parts.
x1 = r1 cos θ1
x1 = (11,648 m)cos(15.95°)
x1 = 11,200 m
x2 = r2 cos θ2
x2 = (8,570 m)cos(11.44°)
x2 = 8400 m
y1 = r1 sin θ1
y1 = (11,648 m)sin(15.95°)
y1 = 3200 m
y2 = r2 sin θ2
y2 = (8,570 m)sin(11.44°)
y2 = 1700 m
Add vectors in the same direction with "ordinary" addition.
x = 11,200 m + 8,400 m
x = 19,600 m
y = 3200 m + 1700 m
y = 4900 m
Using a combination of the pythagorean theorem for magnitude, add vectors at right angles
r = √(x2 + y2)
r = √[(19,600 m)2 + (4,900 m)2]
r = 20,200 m
and tangent for direction.
tan θ = y/x
4900m/19600m
Ques. Find a vector in the direction of vector 
which has a magnitude 21 units. (3 Marks)
Ans. In order to find a vector in the direction of a given vector, first of all we find the unit vector in the direction of direct vector and then multiply it with the given magnitude.
The unit vector in the direction of the given vector \(\overrightarrow {a}\) is
Therefore the vecror f magnitude equal to 21 units and in the direction of is,
For Latest Updates on Upcoming Board Exams, Click Here: https://t.me/class_10_12_board_updates
Check-Out:
Comments