The easiest example of a vector quantity is Force<\/strong>. If you apply force on a body, That force will always have a magnitude<\/mark> (The amount of force you’re putting on the body), as well as the direction<\/mark> (The direction in which you’re applying force on the body).<\/p>\n\n\n\n If a quantity is defined ONLY by its magnitude, that quantity is called a scalar quantity<\/mark>. Time, Volume, Mass are some examples of scalar quantities, they do have magnitude ( example: 5 seconds, 5 liters, 5 kg ), but they don’t have a direction<\/mark>.<\/em><\/p>\n\n\n\n A vector is generally written as a letter with an arrow above it<\/mark> (example: \\( \\vec{A} \\)) or as a Bold letter<\/mark>.<\/p>\n\n\n\n Plane letters are used to denote only the magnitude<\/mark> of the Vector. Magnitude of vector is also denoted with the help of vertical bars on either side of the vector (example: \\(\\left| \\vec A \\right|\\)).<\/p>\n\n\n\n To sum it up,<\/p>\n\n\n\n Graphically, Vectors are represented by a line with an arrowhead at one end.<\/mark> The direction in which the arrowhead points represents the direction of the vector, and the length of the line represents its magnitude.<\/p>\n\n\n Symbol \\(\\hat a\\) in the diagram above is a unit vector, Unit vectors are generally used to indicate the direction of any vector. We will talk more about it in the next section.<\/p>\n<\/div>\n\n\n\n So, If you were applying a force of 10 newtons on a body to push it from left to right, then that force can be represented graphically as:<\/p>\n\n\n\n Any vector whose magnitude is unity ( equal to 1 ) is called a Unit Vector<\/mark>. A unit vector is denoted by a small letter with a hat symbol above it<\/mark> (example: \\( \\hat{a} \\)).<\/p>\n\n\n\n Unit vectors are mostly used to indicate the direction of any vector<\/mark>. Unit vectors are also known as Direction vectors<\/strong><\/mark>.<\/p>\n\n\n\n If a vector \\(\\vec V \\) has magnitude V, and direction parallel to a unit vector \\( \\hat{v} \\), then the vector \\(\\vec V \\) can be written as the product of it’s magnitude and unit vector along it’s direction.<\/p>\n\n\n\n hence, \\( \\vec{V}=V\\ \\hat{v} \\)<\/span><\/p>\n\n\n\n or \\[ \\hat{v}=\\frac{\\vec{V}}{V} \\] <\/span><\/p>\n\n\n\n Note:<\/strong> Unit vectors having direction parallel to X, Y, or Z axis are usually written as \\( \\hat{i},\\hat{j,}\\text{ or }\\hat{k} \\) respectively (Or sometimes, written simply as \\( i,j,\\text{ or }k \\)).<\/p>\n<\/div>\n<\/div><\/div>\n\n\n\n Remember, a vector quantity tells us about only it’s magnitude and direction. It doesn’t tell us anything about the position. This means that a vector doesn’t have a position.<\/mark><\/p>\n\n\n\n So any vector can be drawn anywhere on the graph, regardless of its position.<\/p>\n\n\n\n Or, If we parallel shift a vector from one position to another<\/mark> on the graph without changing its direction and magnitude<\/mark>, the vector remains unchanged!<\/p>\n\n\n For simplicity, It is always a good idea to shift the tail of the vector to the origin.<\/p>\n<\/div><\/div>\n\n\n\n As we already know that a vector \\(\\vec A\\) whose magnitude is A and direction \\(\\hat a\\), holds the relation \\( \\vec{A}=A\\hat{a} \\)<\/p>\n\n\n\n But, we still don’t know what direction is the direction of \\(\\hat a \\)? It could be anything, left to right, top to bottom, or any other direction that we could draw on a paper. That is why, most of the time, vectors are represented with the help of some reference axes.<\/mark><\/p>\n\n\n\n To represent a vector with the help of some reference axes, Any vector \\(\\vec A\\) can be broken down to its multiple component vectors. Each component vector has a direction parallel to one of the reference axis (X, Y, or Z).<\/mark><\/p>\n\n\n\n So the vector \\(\\vec A\\) shown in the figure can be written as:<\/p>\n\n\n\n \\( \\vec{A}=a_1i+a_2j\\) <\/span> (sum of its component vectors)<\/p>\n\n\n\n Note:<\/strong> Only the magnitude part<\/mark> of the “component vector” of a vector is usually called just the “component” of vector. For a vector \\( \\vec A=a_1i+a_2j \\) as shown in figure above, With the help of trigonometric functions, it is clear that<\/p>\n\n\n\n \\( a_1=A\\cos\\theta \\), and Here \\(\\theta\\) is the angle made by the vector with horizontal axis, and A is the magnitude of vector \\(\\vec A\\).<\/p>\n\n\n\n For a vector \\( \\vec A=a_1i+a_2j \\) as shown in figure, With the help of Pythagorean theorem, it is clear that, Resultant vector is the vector resulting from addition (or subtraction) of two or more vectors.<\/p>\n\n\n\n Addition of two vectors is easy!<\/p>\n\n\n\n If we had two vectors \\( \\vec{A}\\text{ and }\\vec{B} \\), such that Then, \\( \\vec{A}+\\vec{B}=A\\hat{a}+B\\hat{b} \\)<\/span> But again, What will be the direction and magnitude of the resultant vector \\(A\\hat{a}+B\\hat{b}\\)?<\/p>\n\n\n\n As usual, to know the direction and magnitude of the resultant vector, Component vectors of vector A and B are required. And there are other methods too for the vector addition, but before we jump to that, let’s first understand how the addition of vectors works in real life with the help of an example.<\/p>\n\n\n\n Consider a body placed on the ground (XY plane)<\/p>\n\n\n\n Let’s now apply a pulling force \\( \\vec{F_x} \\) of magnitude Fx along the direction of X-axis. The body now will start moving in the direction of this force (in the direction of the X-axis).<\/p>\n\n\n\n Or, similarly, If we apply a force \\( \\vec{F_y} \\) of magnitude Fy along the direction of Y axis, The body will move in the direction of Y axis.<\/p>\n\n\n\n Note that the motion of body depends both on direction of force (direction of motion depends on it) as well as on the magnitude of the force (the amount of acceleration depends on it).<\/p>\n\n\n\n But, what if we apply both the forces \\( \\vec{F_x} \\) and \\( \\vec{F_y} \\) on the body at the same time? Obviously, the body won’t move in “only X” or “only Y” direction, It will move in a direction which is a combination of both X and Y directions!<\/p>\n\n\n\n By applying forces \\( \\vec{F_x} \\) and \\( \\vec{F_y} \\) at the same time, we’re actually adding the forces together so that the body moves in the direction of the resultant of these forces.<\/p>\n\n\n\n Now, because the body is moving in a direction different than the direction of forces, It can be assumed that instead of forces \\( \\vec{F_x} \\) and \\( \\vec{F_y} \\), there’s a force of magnitude R acting on the body which makes the body move in its direction<\/p>\n\n\n\n And this force \\( \\vec{R} \\) is actually the resultant of forces \\( \\vec{F_x} \\) and \\( \\vec{F_y} \\)<\/mark>! Note that, the vector R is the resultant of “vector addition” of two forces. It is not equal<\/strong> to the addition of the magnitude of two forces.<\/p>\n\n\n\n For this particular case, since the forces \\( \\vec{F_x} \\) and \\( \\vec{F_y} \\) has the direction along the X and Y axis, It can also be said that the forces \\( \\vec{F_x} \\) and \\( \\vec{F_y} \\) are the component forces<\/u> of the Force \\(\\vec R\\)<\/mark>.<\/p>\n\n\n\n And that’s basically how the vector addition works (Remember, force is a vector quantity). But there comes another question!<\/strong><\/p>\n\n\n\n Why is the Resultant of two vectors A and B drawn like this:<\/p>\n\n\n\n And not like this:<\/p>\n\n\n\n The answer is parallelogram law<\/strong> and triangle law<\/strong> of vectors.<\/p>\n\n\n\n According to the parallelogram law of vectors, If we have two vectors A and B,<\/p>\n\n\n\n And if we parallel shift the two vectors in a way that both the vectors represent adjacent sides of a parallelogram<\/mark>, and, both of the vectors either direct away from or direct towards each other at the same time<\/mark> as shown in the figure:<\/p>\n\n\n\n Then, the diagonal of the parallelogram formed will be the resultant of the two vectors A and B.<\/p>\n\n\n\n <\/p>\n\n\n\n Similar to the Parallelogram law of vectors, we also have triangle law of vectors. If we parallel-shift vector \\(\\vec{B}\\) such that the tail of vector B touches the head of vector A as shown in figure<\/p>\n\n\n <\/p>\n\n\n\n Then, according to the triangle law of vectors,<\/p>\n\n\n\n \\[ \\begin{aligned} &\\vec{R}=\\vec{A}+\\vec{B}\\\\ \\text{or}\\ \\ & \\vec{A}=\\vec{R}-\\vec{B}\\\\ \\text{or}\\ \\ &\\vec{B}=\\vec{R}-\\vec{A} \\end{aligned} \\]<\/span><\/p>\n\n\n\n Now that we know what a vector addition, parallelogram law, and triangle law of vectors are, let’s move on to the types of vector additions.<\/p>\n\n\n\n If \\( \\hat{a}, A\\) and \\( \\hat{b}, B\\) are the direction and magnitude of vectors \\(\\vec A\\) and \\(\\vec B\\) respectively, <\/p>\n\n\n\n such that, \\( \\vec{A}=A\\hat{a} \\) then \\[ \\vec{A}+\\vec{B}=A\\hat{a}+B\\hat{b} \\]<\/span><\/p>\n\n\n\n If both vectors have the same direction, then the resultant of these two vectors, of course, will also be in the same direction, but with the magnitude equal to the sum of the magnitude of the two vectors<\/mark>.<\/p>\n\n\n\n Hence if, \\(\\hat b = \\hat a\\)<\/p>\n\n\n\n then, \\( \\vec{A}+\\vec{B}=\\ A\\hat{a}+B\\hat{a}\\)<\/p>\n\n\n\n or, \\( \\vec{A}+\\vec{B} =\\ \\left(A+B\\right)\\hat{a} \\)<\/span><\/p>\n\n\n\n If \\( A_xi,\\ A_yj \\) and \\( B_xi,\\ B_yj \\) are the component vectors of the vector \\(\\vec A\\), and \\(\\vec B\\) respectively, Then the Resultant of vectors \\( \\vec{A}+\\vec{B} \\) will be equal to the resultant of sum of all their component vectors.<\/p>\n\n\n Hence, if \\( \\vec{A}=A_1i+A_2j \\) Then, \\( \\vec{A}+\\vec{B}=A_1i+A_2j+B_1i+B_2j \\) When we know the magnitude and angle made between two vectors, Parallelogram law can be applied to easily find the direction and magnitude of resultant vector.<\/p>\n\n\n\n Suppose, we had two vectors \\( \\vec{A}\\text{ and } \\vec{B} \\) of magnitude \\[A, B\\] with an angle \u03b8 between them.<\/p>\n\n\n Then, with the help of parallelogram law, And the magnitude of \\(\\vec{R}\\) will be equal to the length of the diagonal of the parallelogram,<\/p>\n\n\n\n \\[ R=\\sqrt{A^2+B^2+2AB\\cos\\theta} \\]<\/span><\/p>\n\n\n\n and, if \\(\\alpha\\) and \\( \\beta \\) are the angle made by resultant R with A and B, then with the help of trigonometry, we have<\/p>\n\n\n\n \\[ \\alpha=\\tan^{-1}\\left[\\frac{B\\sin\\theta}{A+B\\cos\\theta}\\right] \\]<\/span><\/p>\n\n\n\n similarly, \\[ \\beta=\\left(\\theta-\\alpha\\right)=\\tan^{-1}\\left[\\frac{A\\sin\\theta}{B+A\\cos\\theta}\\right] \\]<\/p>\n\n\n\n <\/p>\n\n\n\n Two vectors are said to be orthogonal when they are perpendicular to each other.<\/p>\n\n\n Hence, If the angle between \\( \\vec{A}\\text{ and } \\vec{B} \\) is \\[ 90^{\\circ} \\] and magnitude is A and B respectively, then by putting \\[ \\angle\\theta=90^{\\circ} \\] in above equations,<\/p>\n\n\n\n magnitude, \\( R=\\sqrt{A^2+B^2} \\)<\/span> Just like we can add up two vectors to get a single resultant vector, We can also break (resolve) a single vector into two directions. This is called the Resolution of vector.<\/p>\n\n\n\n If \\(\\vec R\\) is a vector of magnitude R and direction \\(\\hat r\\), and we want to break it into two vectors \\( \\vec{A}\\text{ and }\\vec{B} \\) which make angles \\( \\alpha\\text{ and }\\beta \\) with \\(\\vec R\\) as shown in figure, then the magnitude of vectors \\( \\vec{A}\\text{ and }\\vec{B} \\) can be calculated by applying the opposite of parallelogram law.<\/p>\n\n\n Hence, If \\(\\vec A \\) makes angle \\(\\alpha\\) and \\(\\vec B\\) makes angle \\(\\beta\\) with \\(\\vec R\\)<\/p>\n\n\n\n then, \\[ A=R\\frac{\\sin\\beta}{\\sin\\left(\\alpha+\\beta\\right)} \\]<\/span><\/p>\n\n\n\n and, \\[ B=R\\frac{\\sin\\alpha}{\\sin\\left(\\alpha+\\beta\\right)} \\]<\/span><\/p>\n\n\n\n \\(A\\) and \\(B\\) are magnitude of vectors \\(\\vec{A}\\) and \\(\\vec{B}\\) respectively. And since we know the angles they make with \\(\\vec{R}\\), Their direction is already known.<\/p>\n\n\n\n If \\( \\vec{A}\\text{ and }\\vec{B} \\) are orthogonal, then by putting \\[ \\alpha+\\beta=90^{\\circ} \\] in above two equations (or simply, with the help of trigonometric functions, as we did before to find magnitude of component vectors<\/a>), we get<\/p>\n\n\n\n \\[ A=R\\cos\\alpha\\ =\\ R\\sin\\beta \\] <\/p>\n\n\n\n We’re doing great progress! We know what vectors and their components are, we known how vectors are added to get a resultant vector or broken into their components. On next page, we will talk about product of two vectors.<\/p>\n\n\n\n\n\n\n\n Vector multiplication is a lot different than the simple multiplication (of two scalar quantities). We not only have to think about the multiplication of their magnitudes but also of their directions (Which doesn’t really make sense).<\/p>\n\n\n\n Before we talk about how a vector is multiplied by another vector, let’s first talk about what happens when a scalar quantity (simply a number) is multiplied by a vector quantity.<\/p>\n\n\n\n Multiplication of a vector by a scalar (a number) have two cases:<\/p>\n\n\n\n When a vector is multiplied by a positive number (greater than 0), only the magnitude of the vector gets multiplied by the number, but the direction of vector remains unchanged.<\/p>\n\n\n Thus, if \\(\\vec A\\) is a vector of magnitude A and direction \\(\\hat a\\)<\/p>\n\n\n\n \\( \\vec{A}=A\\hat{a} \\)<\/p>\n\n\n\n Then, the product of vector A with a number \\(n\\) will be given as<\/p>\n\n\n\n \\( n\\times\\vec{A}=n\\times A\\hat{a} \\)<\/p>\n\n\n\n or, \\[ n\\times\\vec{A}=\\left(nA\\right)\\hat{a} \\]<\/span><\/p>\n\n\n\n Note that the \\( \\times \\) symbol here represents a simple multiplication, Don’t confuse it with the “Cross product” which we’ll talk about in upcoming sections.<\/p>\n\n\n\n When a vector is multiplied by a negative number (less than 0), Magnitude of the vector gets multiplied by the number, and the direction of the vector gets reversed<\/mark>.<\/p>\n\n\n\n <\/p>\n\n\n\n Following are some important points to remember when multiplying a vector with a scalar:<\/p>\n\n\n\n And finally, here we are at the last part of this tutorial!<\/p>\n\n\n\n Just think of how a vector (which has both, magnitude as well as direction) can be multiplied by another vector?<\/p>\n\n\n\n You probably may think of many ways that you could put it together, But to get its best use in many circumstances, the product of two vectors is defined in two ways.<\/p>\n\n\n\n (1) The Dot product (also called the scalar product), and (2) The Cross product (also called the vector product)<\/p>\n\n\n\n Dot product<\/strong> is that product of two vectors, In which the magnitude of first vector is multiplied by the magnitude of “Projection of second vector onto the first vector”.<\/p>\n\n\n\n For vector A and B as shown in the image above, Red line represents the projection of vector B onto vector A, and the Blue line represents the Projection of vector A onto vector B<\/p>\n\n\n If we parallel-shift vector A and B in a way so that they meet each other at the endpoint, then, The magnitude of projected lines can be calculated with the help of trigonometric functions.<\/p>\n\n\n\n So if the angle between the two vectors is \u03b8, then<\/p>\n\n\n\n Magnitude of projection of \\(\\vec B\\) onto \\(\\vec A \\ =\\ B\\cos\\theta \\)<\/p>\n\n\n\n similarly, Magnitude of projection of \\(\\vec A\\) onto \\(\\vec B\\ = \\ A\\cos\\theta \\)<\/p>\n<\/div>\n\n\n\n Note:<\/strong> Dot product of two vectors is represented by putting a Dot between the two vectors.<\/p>\n<\/div>\n\n\n\n Hence, If \\(\\vec A\\) and \\(\\vec B\\) are two vectors of magnitude \\(A\\) and \\(B\\)<\/p>\n\n\n Then the Dot product of vector A and B is given as<\/p>\n\n\n\n \\( \\vec{A}\\cdot\\vec{B}=\\left(A\\cos\\theta\\right)B \\) (\\( A\\cos\\theta \\) is magnitude of projection of vector A onto vector B)<\/span><\/p>\n\n\n\n or, \\( \\vec{A}\\cdot\\vec{B}=A\\left(B\\cos\\theta\\right) \\) (\\( B\\cos\\theta \\) is magnitude of projection of vector B onto vector A)<\/span><\/p>\n\n\n\n Or, just simply \\[\\vec{A}\\cdot\\vec{B}=AB\\cos\\theta \\]<\/span> (Equation 1)<\/p>\n\n\n\n <\/p>\n\n\n\n <\/p>\n\n\n\nNotation<\/h3>\n\n\n\n
\n
Graphical representation of vectors<\/h3>\n\n\n\n
![\"Graphical](\"http:\/\/spacican.com\/notes\/wp-content\/uploads\/2020\/09\/drawing-18.png\")
![\"Force](\"http:\/\/spacican.com\/notes\/wp-content\/uploads\/2020\/09\/drawing-1.png\")
Unit Vectors<\/h2>\n\n\n\n
Parallel shifting of vectors<\/h2>\n\n\n\n
![\"Representation](\"http:\/\/spacican.com\/notes\/wp-content\/uploads\/2020\/09\/drawing-2.png\")
Components of vector<\/h2>\n\n\n\n
\n
![\"Components](\"http:\/\/spacican.com\/notes\/wp-content\/uploads\/2020\/08\/drawing-27.png\")
\n
For example, If vector \\( \\vec{A}=a_1i+a_2j \\)
then, \\( a_1i\\text{ and }a_2j \\) are the component vectors of \\(\\vec A\\)
and, \\( a_1\\text{ and }a_2 \\) are the components of \\(\\vec A\\)<\/p>\n<\/div>\n\n\n\nMagnitude of component vectors<\/h3>\n\n\n\n
\\( a_2=A\\sin\\theta \\)<\/span><\/p>\n\n\n\nMagnitude of vector<\/h3>\n\n\n
![\"components](\"http:\/\/spacican.com\/notes\/wp-content\/uploads\/2020\/08\/drawing-26.png\")
Magnitude \\[ A=\\sqrt{a_1^2+a_2^2} \\]<\/span><\/p>\n\n\n\nResultant vector<\/h3>\n\n\n\n
Addition of vectors<\/h2>\n\n\n\n
\\( \\vec{A}=A\\hat{a} \\)
\\( \\vec{B}=B\\hat{b} \\)<\/p>\n\n\n\n
(At this point, you already know what \\( A, B\\text{ and }\\hat{a}, \\hat{b} \\) are)<\/em><\/p>\n\n\n\nExample of vector addition<\/h3>\n\n\n\n
![\"Body](\"http:\/\/spacican.com\/notes\/wp-content\/uploads\/2020\/08\/drawing-28.png\")
![\"Body](\"http:\/\/spacican.com\/notes\/wp-content\/uploads\/2020\/08\/drawing-29.png\")
![\"Body](\"http:\/\/spacican.com\/notes\/wp-content\/uploads\/2020\/08\/drawing-30.png\")
![\"A](\"http:\/\/spacican.com\/notes\/wp-content\/uploads\/2020\/08\/drawing-31.png\")
Hence, \\( \\vec{R}=\\vec{F_x}+\\vec{F_y} \\)<\/p>\n\n\n\n![\"Resultant](\"http:\/\/spacican.com\/notes\/wp-content\/uploads\/2020\/08\/drawing-32.png\")
![\"Resultant](\"http:\/\/spacican.com\/notes\/wp-content\/uploads\/2020\/09\/drawing-3.png\")
![\"Resultant](\"http:\/\/spacican.com\/notes\/wp-content\/uploads\/2020\/09\/drawing-4.png\")
Parallelogram law of vectors<\/h3>\n\n\n\n
![\"two](\"http:\/\/spacican.com\/notes\/wp-content\/uploads\/2020\/09\/drawing-5.png\")
![\"Vectors](\"http:\/\/spacican.com\/notes\/wp-content\/uploads\/2020\/09\/drawing-6.png\")
![\"Parallelogram](\"http:\/\/spacican.com\/notes\/wp-content\/uploads\/2020\/09\/drawing-7.png\")
Triangle law<\/h3>\n\n\n\n
![\"Triangle](\"http:\/\/spacican.com\/notes\/wp-content\/uploads\/2020\/09\/drawing-31.png\")
Addition of two vectors when their direction and magnitude is known<\/h2>\n\n\n\n
and \\( \\vec{B}=B\\hat{b} \\)<\/p>\n\n\n\nAddition of two vectors having the same direction<\/h3>\n\n\n\n
Addition of Two vectors when their components are known<\/h2>\n\n\n\n
![\"Resultant](\"http:\/\/spacican.com\/notes\/wp-content\/uploads\/2020\/09\/drawing-8.png\")
and \\( \\vec{B}=B_1i+B_2j \\)<\/p>\n\n\n\n
or, \\[ \\vec{A}+\\vec{B}=\\left(A_1+B_1\\right)i+\\left(A_2+B_2\\right)j \\]<\/span><\/p>\n\n\n\n\n
Addition of two vectors when magnitude and angle between them is known<\/h2>\n\n\n\n
![\"Resultant](\"http:\/\/spacican.com\/notes\/wp-content\/uploads\/2020\/09\/drawing-9.png\")
\\( \\vec{R}=\\vec{A}+\\vec{B} \\)<\/p>\n\n\n\nResultant of two orthogonal vectors<\/h3>\n\n\n\n
![\"Resultant](\"http:\/\/spacican.com\/notes\/wp-content\/uploads\/2020\/09\/drawing-10.png\")
and angle, \\[ \\alpha=\\tan^{-1}\\left[\\frac{B}{A}\\right] \\]<\/span><\/p>\n\n\n\n\n
Resolution of a vector in two directions<\/h2>\n\n\n\n
![\"Resolution](\"http:\/\/spacican.com\/notes\/wp-content\/uploads\/2020\/09\/drawing-11.png\")
Resolution of a vector into two orthogonal vectors<\/h3>\n\n\n
![\"Resolution](\"http:\/\/spacican.com\/notes\/wp-content\/uploads\/2020\/09\/drawing-12.png\")
\\[ B=R\\sin\\alpha\\ =\\ R\\cos\\beta \\]<\/span><\/p>\n\n\n\nProduct of Vectors<\/h2>\n\n\n\n
Product of a vector and a scalar<\/h3>\n\n\n\n
Multiplication of Vector with a Positive number<\/h4>\n\n\n\n
![\"Multiplication](\"http:\/\/spacican.com\/notes\/wp-content\/uploads\/2020\/09\/drawing-13.png\")
Multiplication of vector with a negative number<\/h4>\n\n\n
![\"Multiplication](\"http:\/\/spacican.com\/notes\/wp-content\/uploads\/2020\/09\/drawing-14.png\")
\n
\\[ \\begin{aligned} &n(\\vec{A}+\\vec{B}) = n\\vec{A}+n\\vec{B} \\\\ \\text{and, }\\ &n_1\\vec{A}+n_2\\vec{A} = \\left(n_1+n_2\\right)\\vec{A} \\end{aligned} \\]<\/span><\/li>\n\n\n\n
\\( n\\vec{A}=\\vec{A}n \\)<\/span><\/li>\n\n\n\n
\\( 1\\times\\vec{A}=\\vec{A} \\)<\/span><\/li>\n\n\n\n
\\( 0\\times\\vec{A}=\\vec{0} \\)
The magnitude of components of a zero vector is 0
\\( \\vec{0}=0i+0j+0k \\)<\/em><\/span><\/li>\n<\/ul>\n\n\n\nProduct of Vector with Vector<\/h3>\n\n\n\n
Dot Product (Scalar Product)<\/h2>\n\n\n\n
![\"Projection](\"http:\/\/spacican.com\/notes\/wp-content\/uploads\/2020\/09\/drawing-16.png\")
![\"Magnitude](\"http:\/\/spacican.com\/notes\/wp-content\/uploads\/2020\/09\/drawing-17.png\")
![\"Scalar](\"http:\/\/spacican.com\/notes\/wp-content\/uploads\/2020\/08\/drawing-17.png\")
\n
![\"Orthogonal](\"http:\/\/spacican.com\/notes\/wp-content\/uploads\/2020\/09\/drawing-19.png\")
Or alternatively, by putting \\[ \\theta=90^{\\circ} \\] in equation 1<\/a>,
\\[ \\vec{A}\\cdot\\vec{B}=AB\\cos90^{\\circ}=0 \\]<\/span><\/li>\n\n\n\n
then, \\( i\\ \\perp\\ j\\ \\perp\\ k \\)
hence, \\[ i\\cdot j=j\\cdot k=k\\cdot i=0 \\]<\/span><\/span><\/li>\n\n\n\n
\\[Ai\\cdot Bi=AB\\cos0^{\\circ}=AB \\]<\/span>
Dot product of two unit vectors having same direction is unity (equal to one)
\\( \\hat{a}\\cdot\\hat{a}=1 \\)
similarly, \\( i\\cdot i=j\\cdot j=k\\cdot k=1 \\)<\/span><\/span><\/li>\n\n\n\n
\\( \\vec{A}\\cdot\\vec{B}=\\vec{B}\\cdot\\vec{A} \\)<\/span><\/li>\n\n\n\n
\\( \\vec{A}\\cdot\\vec{B}\\cdot\\vec{C}=\\vec{A}\\cdot\\left(\\vec{B}\\cdot\\vec{C}\\right)=\\left(\\vec{A}\\cdot\\vec{B}\\right)\\cdot\\vec{C} \\)<\/span><\/li>\n\n\n\n