SHORT NOTES: Vectors

Scalars only magnitude.Ex: Distance,Temperature, speed, mass

Vectors both magnitude and direction. The magnitude of $\vec{v}$ is written $\left |\vec{v} \right |\equiv v$ .Ex: displacement, velocity, acceleration and force

Unit Vector: only give direction i.e vector with magnitude "one".

Ex: $\frac{\vec{v}}{v}=\hat{v}$ (unit vector)

Note: $\hat{i},\hat{j},\hat{k}$ are unit vectors along x,y,z axis.

Vectors are equal if they have the same magnitude and direction.

The negative of a vector has the same magnitude but opposite direction.

Scalar multiplication:Multiplication or division of a vector by a scalar results in a vector for which

(a)only the magnitude changes if the scalar is positive

(b)the magnitude changes and the direction is reversed if the scalar is negative.

The projections of a vector along the axes of a rectangular co-ordinate system are called the components of the vector.

$\vec{A}=A_x\hat{i}+A_y\hat{j}$

Similarly for 3D i.e for x,y,z axis.

Addition of vectors:

1.To add vectors by components $\vec{R}=\vec{A}+\vec{B}$

Find $R_x=A_x+B_X$ and $R_y=A_y+B_y$ .

Then $\left |\vec{R} \right |={\sqrt{R_x^2+R_y^2}}$ and $\theta =\tan^{-1}\frac{R_y}{R_x}$

2.Triangle Law: $R^2=A^2+B^2+2AB\cos \theta$ and $\tan \beta =\frac{B\sin \theta}{A+B\cos \theta}$

$\theta$ is angle between A and B vector i.e by how much angle the vector B rotates/shift from the direction of vector A in anti-clockwise direction.

$\beta$ is angle between A and R vector

3.Parallelogram: Same formula as triangle law

Subtraction of a vector is defined by adding a negative vector.

Dot Product:

$\vec{A}.\vec{B}=AB\cos \theta$ Given A , B , $\theta$ are known

$\vec{A}.\vec{B}=A_xB_x+A_yB_y+A_zB_z$ If A,B are expressed in compent form

Note: $\theta=\cos^{-1}\frac{A_xB_x+A_yB_y+A_zB_z}{AB}$ A,B are magnitude

Geometrical interpretations: $\vec{A}.\vec{B}=AB\cos \theta$ $=A\cos \theta X B$ = Projection of A on B

Cross products:

$\vec{A}x\vec{B}=AB\sin \theta\hat{n}$ where $\hat{n}$ is the normal to the plane containing A & B with direction defined by thumb rule or screw rule

$\vec{A}x\vec{B}=\begin{vmatrix} \hat{i} &\hat{j} &\hat{k} \\A_X &A_y &A_z \\B_x &B_y &B_z \end{vmatrix}$

Note:

$\begin{align} \mathbf{i}&=\mathbf{j}\times\mathbf{k}\\ \mathbf{j}&=\mathbf{k}\times\mathbf{i}\\ \mathbf{k}&=\mathbf{i}\times\mathbf{j} \end{align}$ $\begin{align} \mathbf{k\times j}=&-\mathbf{i}\\ \mathbf{i\times k}=&-\mathbf{j}\\ \mathbf{j\times i}=&-\mathbf{k} \end{align}$ $\mathbf{i}\times\mathbf{i}=\mathbf{j}\times\mathbf{j}=\mathbf{k}\times\mathbf{k}=\mathbf{0}$

Geometrical interpretations:

The magnitude of the cross product can be interpreted as the positive area of the parallelogram having a and b as sides

$A = \left\| \mathbf{a} \times \mathbf{b} \right\| = \left\| \mathbf{a} \right\| \left\| \mathbf{b} \right\| \sin \theta. \,\!$

Similarly Area of triangle is half of cross product.

Scalar triple products:

The scalar triple product of three vectors A, B, and C is denoted [A,B,C] and defined by

Geometric interpretation:

Geometrically, the scalar triple product

$\mathbf{a}\cdot(\mathbf{b}\times \mathbf{c})$

is the (signed) volume of the parallelepiped

$\mathbf{a}\cdot(\mathbf{b}\times \mathbf{c}) = \det \begin{bmatrix} a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \\ c_1 & c_2 & c_3 \\ \end{bmatrix}.$

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Vectors

Geometric interpretation:

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