Exams Prep Master
A vector is known as a quantity that has both; magnitude and direction but not a position. For example; velocity and acceleration. In geometry, a vector is a directed line segment, whose length is the magnitude of the vector and with an arrow that indicates the direction. Typically, the vector is represented by an arrow whose direction is similar to that of the quantity and whose length is proportional to the quantity magnitude. Moreover, vector quantities are generally represented by scaled vector diagrams. The vector diagram signifies a displacement vector. Here, we will learn more about vector formulas, triangle and parallelogram law of addition, and discuss a few important questions.
Read Also: Equation Line
| Table of Content |
Key Terms: Scalar, Triple, Product, Vectors, magnitude of this vector, Dot and Cross product, Resultant, vector formulas, geometry, vector quantities, parallelogram law of addition
What is Vector Formula?
[Click Here for Sample Questions]
As mentioned above, the vector is basically represented by the arrow, whose direction is the same as the quantity and whose length is directly proportional to its magnitude. The vector diagram will depict a displacement vector:
- The mathematical representation of physical quantities of both: the magnitude and the direction are termed as vectors.
- Vectors are those which are added geometrically and not algebraically,
- Their resultant has to be calculated independently,
- The magnitude of this vector is given as |ab| or |a|. It represents the vector length & with the help of the Pythagorean theorem,
- Vector of any physical quantity is represented as a straight line with an arrowhead, the length of the straight line denotes the magnitude of the vector and the arrowhead gives its direction.
- For better understanding, a vector is the force applied to an object because both the strength and direction of the applied force affect its action on the object.
Vector
Check More: Angle Between a Line and a Plane
Triangular Law of Addition
[Click Here for Sample Questions]
The vector addition is done on the basis of triangle law. If both forces vector a and vector b acts in the same direction, then its resultant vector r will be the sum of two vectors.
Triangular Law of Addition
The formula for triangular law of addition: vector (r =a + b)
Read Further: Three Dimensional Geometry
Parallelogram Law of Addition
[Click Here for Sample Questions]
If the two forces are represented vector a and vector b by the adjacent sides in the parallelogram, then the result will be represented by the diagonal of a parallelogram drawn from the same point.
Parallelogram Law of Addition
The formula for parallelogram as the law of addition is: vector (r = a + b)
Polygon Law of Vector
[Click Here for Sample Questions]
When the numbers of vectors are represented as magnitude & direction, then their resultant is represented, such that the closing side of the polygon is taken in the opposite direction,
The formula is: ab + bc +cd + de = ae. Ae = closing side
Also Read: Angle Between Two Planes
Types of Vectors
[Click Here for Sample Questions]
Vectors are of various types:
- Zero vector or null vector- When the magnitude of the vector is zero or we can say that the starting and the endpoint of that vector are the same, then a vector is said to be a zero vector. The zero vectors have no specific direction.
Zero vector or null vector
For example: ab is a line segment the coordinates of point a is the same as that of point b. So, the vector is denoted by 0.
- Unit vector- When the magnitude of the vector is one unit in length, then a vector is said to be a Unit Vector. But two vectors may not be equal as they have different directions.
For example: if p is a vector having a magnitude p and has the magnitude equal to 1.
Check out Important Notes for Types of Vectors
- Position vector- When a vector denotes the position of a point with respect to its origin, then the vector is said to be a position vector.
Position vector
- Co-initial vectors- When two or more vectors have the same starting point, then the vector is said to be co-initial vectors. For example, two vectors, ab, and ac are called co-initial vectors because they have the same starting point a.
- Like and unlike vectors- The vectors having the same directions are said to be like vectors whereas vectors having opposite directions are said to be unlike vectors.
- Co-planar vectors- Three or more vectors lying in the same Plane are known as coplanar vectors.
- Collinear vectors- Vectors which are also known as parallel vectors are those vectors that lie the same line with respect to their magnitude and direction.
- Equal vectors- Two vectors are said to be equal vectors when they have both direction and magnitude equal, even if they have different initial points.
- Displacement vectors- The vector ab represents a displacement vector if a point is displaced from position a to b.
- Negative vectors- Negative vectors are those whose points are lied in the opposite direction from the positive direction.
- Euclidean vector- Any geometric object which has both magnitude and direction is called a Euclidean vector.
The video below explains this:
Types of Vectors Detailed Video Explanation:
Also Read:
| Related Articles | ||
|---|---|---|
| Vector Product | Position Vector | Vector Addition |
| Negative Vector | Angle between two vectors | Properties of Determinant |
Mathematical Operations on Vector
[Click Here for Sample Questions]
- Vector addition-
The parallelogram law gives the rule for the addition of two or more vectors, according to this law, the two vectors are added, which are represented by two sides of a triangle with the same magnitude and direction & the third side gives of the resultant addition vector. The vector sum is obtained by placing those head to tail.
- Vector subtraction-
Subtraction of vector means the Addition of Vectors with negative of another vector,
For example: there are two vectors q and r. If vector ‘q’ is to be subtracted from vector ‘r’, then the negative of vector ‘q’ should be found and it should be added to vector ‘r’ using triangle law. Subtracting vectors is not commutative
It means the formula lies here are: r - q = r + (-q).
Check Important Notes for Difference between Scalar and Vector Quantities
- Vector multiplication-
It refers to the technique for the multiplication of two or more vectors with each other. When two arbitrary vectors are multiplied, the magnitude of the number is different, but the scalar product is similar.
formula- a*b = ll a ll ll b ll sin \(\Theta\)n
| Ll a ll | = | Length of vector a |
| Ll b ll | = | Length of vector b |
| \(\Theta\) | = | The angle between a and b |
| N | = | Unit vector perpendicular to the plane containing a and b |
Things to Remember
- The vector is typically represented by the arrow, whose direction is the same as that of the quantity and whose length is directly proportional to its magnitude. The vector diagram will depict a displacement vector.
- The vector addition is done based on the triangle law. If both forces vector a and vector b acts in the same direction, then its resultant vector r will be the sum of two vectors.
- If the two forces are represented vector a and vector b by the adjacent sides in the parallelogram, then the result will be represented by the diagonal of a parallelogram drawn from the same point.
- When the numbers of vectors are represented as magnitude & direction, then their resultant is represented, such that the closing side of the polygon is taken in the opposite direction.
- When the magnitude of the vector is zero or we can say that the starting and the endpoint of that vector are the same, then a vector is said to be a zero vector. The zero vectors have no specific direction.
Also Read:
Sample Questions
Ques: Give the vector for each of the following: (3 marks)
a) The vector from (2, -7, 0) to (1, -3, -5)
b) The vector from (1, -3, -5) to (2, -7, 0)
c) The position vector for (-90, 4)
Ans:
- To construct this vector we need to subtract coordinates of the starting point from the ending point.
{1 -2, -3 – (-7), -5 -0} = {-1, 4, -5}
- Same thing is repeated here: {2 -1, -7 – (-3), 0 (-5)} = {1, -4, 5}
It is to be noted that the only difference between the first two is the signs are all opposite. This difference is significant as it is this difference that signifies the two vectors point in opposite directions.
- Here the position vector of a point is nothing more than a vector with the point’s coordinates as its component. (-90, 4)
Ques: If ABCDEF is a regular hexagon, then vector AD + vector EB + vector FC equals. (3 marks)
a) \(2 \overrightarrow{AB}\)
b) \(\overrightarrow{O}\)
c) \(3 \overrightarrow{AB}\)
d) \(4 \overrightarrow{AB}\)
Ans: The correct answer is D.
Ques: If ABCDEF is a regular hexagon, then what is the value of vector (AD + EB + FC)? (2 marks)
Ans:
We have,
Ques: Five forces represented by the vector (AB, AC, AD), vector AE and vector AF act at the vertex A of a regular hexagon ABCDEF. Prove that their result is a force represented by 6 vector AO, where O is the centre of the hexagon. (3 marks)
Ans:
This is the required answer.
Ques: If vector (a, b, c, d) are the position vectors of points A, B, C, D such that no three of them are collinear and vector (a + b + c = b + d), then a, b, c, d is: (4 marks)
a) Rhombus
b) Rectangle
c) Parallelogram
d) Square
Ans: The correct answer is C. parallelogram.
Multiplying the above equation by ½
\(\frac{1}{2}(\overrightarrow{a}+\overrightarrow{c}) = \frac{1}{2}(\overrightarrow{b}+\overrightarrow{d})\)
Therefore, the position vector of the midpoint of BD = Position Vector of the midpoint of AC. Hence the diagonals bisect each other and the given figure ABCD is a parallelogram.
Ques: If the vectors (a, b, c) are three non-zero vectors, no two of which are collinear and the vectors (a + b) is collinear with vector c, and vectors (b + c) is collinear with vector (a), then vectors (a, b, c): (4 marks)
a) Vector a
b) Vector b
c) Vector c
d) None of these
Ans: The correct answer is D.
As vectors (a + b) is collinear with vector c,
\(\therefore \overrightarrow{a} + \overrightarrow{b} = \lambda \overrightarrow{c}........(i)\)
As vectors (b + c) is collinear with vector (a)
\(\therefore \overrightarrow{b} + \overrightarrow{c} = \mu \overrightarrow{a}........(ii)\)
Adding vector c to both sides of the equation (i)
\(\overrightarrow{a} + \overrightarrow{b} + \overrightarrow{c} = \lambda \overrightarrow{c} + \overrightarrow{c}\)
\(\overrightarrow{a} + \overrightarrow{b} + \overrightarrow{c} = \overrightarrow{c}(\lambda + 1)........(iii)\)
Adding vector a to both sides of the equation (iii)
\(\overrightarrow{a} + \overrightarrow{b} + \overrightarrow{c} = \mu \overrightarrow{a} + \overrightarrow{a}\)
\(\overrightarrow{a} + \overrightarrow{b} + \overrightarrow{c} = \overrightarrow{a}(\mu + 1)........(iv)\)
Equating the RHS of the equations (iii) and (iv),
\(\overrightarrow{c}(\lambda + 1) = \overrightarrow{a}(\mu + 1)\)
As a is not collinear with c,
\(\therefore \lambda + 1 = \mu + 1 = 0\)
\(\overrightarrow{a} + \overrightarrow{b} + \overrightarrow{c} = \text 0 \text {{From equation (iv)}}\)
\(\therefore \overrightarrow{a} + \overrightarrow{b} + \overrightarrow{c} = 0 \)
For Latest Updates on Upcoming Board Exams, Click Here: https://t.me/class_10_12_board_updates
Check-Out:
Comments