The unit vector along \\(\hat{i} + \hat {j}\\) is:

Unit vector along\\( ( i + j) is\dfrac {( i + j)}{| i + j|}\\) \\(|i + j | \ is \ magnitude \ of ( i + j)\\) we know, i and j are perpendicular to each other So, \\(|i + j| = \sqrt{i^2 + j^2 + 2 i j cos 90^o}\\)\\(\sqrt{ 1 + 1} = \sqrt{2}\\)hence, unit vector along \\((i + j) is \dfrac{(i + j)}{√2}\\)

The magnitude of vectors \\(\vec{A}\\), \\(\vec{B}\\) and \\(\vec{C}\\) are \\(3\\), \\(4\\) and \\(5\\) units respectively. if \\(\vec{A}+\vec{B} = \vec{C}\\), find the angle between \\(\vec{A}\\) and \\(\vec{B}\\).

\\(\textbf{Step 1:}\\) \\(\textbf{ Squaring the given equation on both sides}\\)Let \\(\theta\\) be the angle between \\(\vec{A}\\) and \\(\vec{B}\\) \\(\vec{A} + \vec{B} = \vec{C}\\)Squaring on both sides \\((\vec{A} + \vec{B})^2 = (\vec{C})^2\\) \\(\Rightarrow|\vec{A}|^2 + |\vec{B}|^2 + 2 \vec{A}. \vec{B} = |\vec{C}|^2\\) \\(\Rightarrow |\vec{A}|^2 + |\vec{B}|^2 + 2 |\vec{A}||\vec{B}| \cos \theta = |\vec{C}|^2\\) \\(....(1)\\)\\(\textbf{Step 2:}\\) \\(\textbf{Calculations}\\)Put the values of all the variables in eq \\((1)\\) to get \\(\theta\\) \\((3)^2 + (4)^2 + 2 \times 3 \times 4 \, \cos \theta = (5)^2\\) \\(\Rightarrow\\) \\(\cos \theta = 0\\) \\(\Rightarrow \\) \\(\theta = 90^{\circ}\\)

Which of the following is vector quantity?

A) Speed is a scalar quantity, having only magnitude and giving no information about direction. For example, 40 m\s is a speed, it tells us how fast an object is traveling, but nothing abut which direction the object is traveling in. The vector counterpart to speed is velocity.B) Displacement is a vector quantity because it has both magnitude and direction.C) Mass is a scalar quantity. Simply put, it refers to how much matter an object is made up of. It has magnitude but gives no indication of direction in any sense. The vector counterpart to mass is weight.D) Time is a scalar quantity (as far as we are concerned at this level). It gives information about magnitude, i.e. how much time, but no information about direction.

Which of the following is not a vector quantity?

Distance is the quantity which is not vector.

Which one of the following is a scalar quantity?

Work is a scalar quantity because it is the dot product of two vectors (Force and displacement). All the others have direction and magnitude so they are vectors.

A vector is not changed if

A vector has both magnitude and direction. A vector is changed if its rotated through certain angle as its direction has been changed. if vector is multiplied by scalar quantity its magnitude is changed. a vector does not changed if it is displaced parallel to itself.

If a particle moves from points \\(P(2, 3, 5)\\) to point \\(Q (3, 4, 5)\\). Its displacement vector be:

If a particle moves from \\(P(2,3,5)=2\hat{i}+3\hat{j}+5\hat{k}\\Q(3,4,5)=3\hat{i}+4\hat{j}+5\hat{k}\\)Now the distance between them\\(=(3\hat{i}+4\hat{j}+5\hat{k})-(2\hat{i}+3\hat{j}+5\hat{k})\\=\hat{i}+\hat{j}\\)

Define scalars and vectors. Give two examples of each.

Scalars are the physical quantities that have the only magnitude. The examples of scalars are electric charge, density, mass etc. Vectors are the physical quantities that have both magnitudes as well as direction. The examples of vectors are velocity, acceleration, force etc.

What is the angle between (A+B) and (A x B)? Give reason.

Given, Two vectors A and B.A and B are two vectors. So, their sum \\(A +B\\) lies in the same plane where A and B lie (Since they are non -parallel so they define a plane and the cross product between them is not zero.)\\(A \times B =| A ||B| sinα n\\), where α is the angle between A & B and n is the unit vector perpendicular to the plane containing A & B. So, the angle between \\((A+B)\\) and \\((A \times B)\\) is \\(90^0.\\)Mathematically,\\(|A +B||A\times B|cosα=(A+B).(A\times B)\\)\\(= A.(A\times B)+B.(A\times B)\\)\\(= B.(A\times A)+A.(B\times B)\\)\\(= 0+0\\)

The magnitude of a given vector with end points (4, -4, 0) and (-2, -2, 0) must be:

\\(\vec r=\vec r_2 -\vec r_1 =(-2\hat i- 2\hat j +0\hat k)-(4\hat i -4\hat j +0\hat k)\\)\\(\Rightarrow \vec r =-6\hat i+2\hat j+0\hat k\\)\\(\therefore |\vec r|=\sqrt {(-6)^2 +(2)^2 +0^2} =\sqrt {36+4}=\sqrt {40}=2\sqrt {10}\\).

Scalar and Vector Quantities | Definition, Examples, Diagrams

definition

Scalar Quantities

A scalar quantity is a quantity which is defined by only magnitude. Some examples of scalar quantities are Mass, Charge, Pressure, etc.

definition

Vector quantities

Vector quantities are those which have both magnitude and direction and obey vector laws of addition. Some examples of vectors are displacement, velocity, force, etc.
A quantity is called a vector only if it follows all the above three conditions. For example, current is not a vector despite having both magnitude and direction because it does not follow vector laws of addition.

result

Scalars and Vectors

Following are some differences listed between scalars and vectors.

S.No.	Scalars	Vectors
1.	Have only magnitude	Have both magnitude and direction
2.	Algebra: Same as real numbers	Algebra: Follow vector laws of addition
3.	Examples: Mass, charge, etc	Examples: velocity, force, electric field. etc.

definition

Scalar

Scalar Quantities:
1. A physical quantity which is having magnitude only but not direction those are scalar quantities.
2. These are one-dimensional quantities.
3. It follows ordinary rules of algebra.
4. The scalar can be divided by any other scalar quantity.
5. It changes due to change in their magnitude only.
6. e.g. mass, work etc

definition

vector

Vector Quantity:
1. A vector quantity is one, that has both magnitude and direction.
2. Are multi-dimensional quantities.
3. It changes with the change in their direction or magnitude or both.
4. Follow rules of vector algebra.
5. Two vectors can never divide.

diagram

Direction of vector

Vector

a

shown in diagram can be represented as

a = a_{x} \hat{i} + a_{y} \hat{j} + a_{z} \hat{k}

Magnitude of vector a
a =

a_{x}^{2} + a_{y}^{2} + a_{z}^{2}

a_{x}

- component of vector along x-axis

a_{y}

- component of vector along y-axis

a_{z}

- component of vector along z- axis

\hat{i}, \hat{j} a n d \hat{k}

- unit vectors along x,y and z direction.

Scalar and Vector Quantities

definition

Scalar Quantities

definition

Vector quantities

result

Scalars and Vectors

definition

Scalar

definition

vector

diagram

Direction of vector

REVISE WITH CONCEPTS

Equality of Vectors

Quick Summary With Stories

Scalar and Vector Quantities with Simple Examples

The unit vector along $\hat{i} + \hat{j}$ is:

The magnitude of vectors $A$ , $B$ and $C$ are $3$ , $4$ and $5$ units respectively. if $A + B = C$ , find the angle between $A$ and $B$ .

Which of the following is vector quantity?

Which of the following is not a vector quantity?

Which one of the following is a scalar quantity?

A vector is not changed if

If a particle moves from points $P (2, 3, 5)$ to point $Q (3, 4, 5)$ . Its displacement vector be:

Define scalars and vectors. Give two examples of each.

What is the angle between (A+B) and (A x B)? Give reason.

The magnitude of a given vector with end points (4, -4, 0) and (-2, -2, 0) must be:

definition

Scalar Quantities

definition

Vector quantities

result

Scalars and Vectors

definition

Scalar

definition

vector

diagram

Direction of vector

REVISE WITH CONCEPTS

Equality of Vectors

Quick Summary With Stories

Scalar and Vector Quantities with Simple Examples

The unit vector along i^+j^​ is:

The magnitude of vectors A, B and C are 3, 4 and 5 units respectively. if A+B=C, find the angle between A and B.

Which of the following is vector quantity?

Which of the following is not a vector quantity?

Which one of the following is a scalar quantity?

A vector is not changed if

If a particle moves from points P(2,3,5) to point Q(3,4,5). Its displacement vector be:

Define scalars and vectors. Give two examples of each.

What is the angle between (A+B) and (A x B)? Give reason.

The magnitude of a given vector with end points (4, -4, 0) and (-2, -2, 0) must be:

The unit vector along $\hat{i} + \hat{j}$ is:

The magnitude of vectors $A$ , $B$ and $C$ are $3$ , $4$ and $5$ units respectively. if $A + B = C$ , find the angle between $A$ and $B$ .

If a particle moves from points $P (2, 3, 5)$ to point $Q (3, 4, 5)$ . Its displacement vector be: