Revision: Vectors and Three-dimensional Geometry >> Vectors Maths Commerce (English Medium) Class 12 CBSE

In three-dimensional geometry, the vector drawn from the origin O(0, 0, 0) to a point P(x, y, z) is called the position vector of the point P. It is written as \[\vec{OP}\]. If point P(x, y, z) is given, then the magnitude of its position vector is:

A vector quantity is a physical quantity that has magnitude as well as direction.

A scalar quantity is a physical quantity that has magnitude only.

A vector is a quantity that has magnitude as well as direction. Geometrically, a vector is represented by a directed line segment such as  \[\vec{AB}\], where A is the initial point and B is the terminal point.

The magnitude of vector \[\vec{AB}\] is the length of the directed line segment AB. It is written as \[|\vec{AB}|\], \[|\vec{a}|\], or simply a. The magnitude of a vector is never negative because it represents length.

If l, m, n are direction cosines of a line and if a, b, c are real numbers such that \[\frac{\mathrm{a}}{l}=\frac{\mathrm{b}}{\mathrm{m}}=\frac{\mathrm{c}}{\mathrm{n}}=\lambda,\] then a, b, c are called direction ratios of that line.

The angles made by a vector with the positive directions of the X-axis, Y-axis and Z-axis are called direction angles of the vector, denoted by α, β, and γ.

If α, β and γ are the direction angles of a vector, then the cosines of these angles, i.e.

l = cos⁡α, m = cos⁡β, n = cos⁡γ 

are called the direction cosines of the vector.

If point is (x,y,z) and distance r: \[\cos\alpha=\frac{x}{r},\quad\cos\beta=\frac{y}{r},\quad\cos\gamma=\frac{z}{r}\]

If \[\vec{a}\] is a vector and \[\lambda\] is a scalar, then \[\lambda\vec{a}\] is called the multiplication of the vector \[\vec{a}\] by the scalar \[\lambda\]. The resulting quantity is also a vector, and it is collinear with \[\vec{a}\].

If ā and b̄ are any two vectors, then the scalar product of these vectors is

ā · b̄ = |ā| |b̄| cos θ = ab cos θ, where θ is the angle between ā and b̄.

When the direction of rotation is anticlockwise, then the rotation will move the screw upwards. It is called a right-handed orientation or a right-handed screw rule. 

Two vectors \[\vec{a}\] and \[\vec{b}\] are parallel if one is a scalar multiple of the other.

\[\vec{a}=\lambda\vec{b}\] (λ is a scalar)

The cosines of the angles made by a vector with the positive directions of the coordinate axes are called the direction cosines of the vector.

If a vector \[\vec{a}\] makes angles α,β,γ with the positive x, y and z axes respectively, then:

l = cosα, m = cosβ, n = cosγ

are called the direction cosines of the vector.

In Cartesian Form:

\[l=\frac{x}{r},\quad m=\frac{y}{r},\quad n=\frac{z}{r}\]

For any three given vectors, the scalar product of one of the vectors and the cross product of the remaining two, is called a scalar triple product

Thus, \[\vec{a},\vec{b},\vec{c}\] are three vectors, then \[(\vec{a}\times\vec{b})\cdot\vec{c}\]is called the scalar triple product and is denoted by \[[\vec{a}\vec{b}\vec{c}]\mathrm{~or~}[a,b,c]\]

Any three numbers proportional to direction cosines are called direction ratios.

• Denoted by a, b, c

• A line has infinitely many direction ratios.

\[l=\frac{a}{\sqrt{a^2+b^2+c^2}}\], \[m=\frac{b}{\sqrt{a^2+b^2+c^2}}\], \[n=\frac{c}{\sqrt{a^2+b^2+c^2}}\]

Let\[\vec{a}\] and \[\vec{b}\]be two non-zero, non-parallel vectors, and let θ be the angle between them such that (0 < θ < π).

\[\vec{a}\times\vec{b}=\left|\vec{a}\right|\left|\vec{b}\right|\sin\theta\left.\hat{n}\right.\]

or

\[\vec{a}\times\vec{b}=ab\sin\theta\mathrm{~}\hat{n}\]

where \[\hat{n}\] is a unit vector perpendicular to both \[\vec{a}\] and\[\vec{b}\] such that\[\vec{a}\], \[\vec{b}\], \[\hat{n}\] form a righthanded triad of vectors.

 The square of a vector a, i.e., \[\vec{a^2}\] is a scalar which denotes the square of the length of a and is equal to the square of its modulus.

\[\vec{a^2}\] = \[|\vec{a}|^2\] 

A directed line segment is a line segment with an arrowhead showing direction. Its two endpoints are distinguishable as the initial point and the terminal point

The vector is denoted by \[\overrightarrow{AB}\]  

The vector drawn from the origin O(0,0,0)to a point P(x,y,z) is called the position vector of the point P.

It is denoted by: \[\vec{OP}=x\hat{i}+y\hat{j}+z\hat{k}\]

Magnitude of Position Vector: \[|\vec{OP}|=\sqrt{x^2+y^2+z^2}\]

When quantities can be represented by a certain number of units with no association with direction in space, they are called scalar quantities and numbers that represent them are called scalars. 

The scalar product or inner product of two non-zero vectors written as like \[\mid a\mid\mid b\mid\cos\theta\]\[\vec{a}\], \[\vec{b}\] is defined to be the scalar \[\left|\vec{a}\right|\left|\vec{b}\right|\cos\theta\] = \[ab\cos\theta\]

where a \[=|\vec{a}|\], b = \[=|\vec{b}|\] and θ = (0 ≤ θ ≤ π) is the angle between\[\vec{a}\] and \[\vec{b}\].

Let \[\hat{i}\],\[\hat{j}\], \[\hat{k}\] be unit vectors in the positive direction of the three mutually perpendicular coordinate axes, x-axis,  y-axis and z-axis, respectively. Then, these vectors are said to form an orthonormal triad of vectors. 

Dot Products: 

• \[\hat{i}\cdot\hat{i}=\hat{j}\cdot\hat{j}=\hat{k}\cdot\hat{k}=1\]

• \[\hat{i}\cdot\hat{j}=\hat{j}\cdot\hat{k}=\hat{k}\cdot\hat{i}=0\]

If a vector \[\overrightarrow{AB}\] is denoted by \[\overrightarrow{a}\], then \[\mid\overrightarrow{a}\mid\] denotes the positive length of the vector a, also called the magnitude or norm or modulus of the vector.

Thus \[\left|\vec{a}\right|\] = a, if a is the positive length of \[\overrightarrow{a}\].

\[\mid\overset{\rightarrow}{\operatorname*{\mathbf{AB}}}\mid=\mid\overset{\rightarrow}{\operatorname*{a}}\mid=a\]

A quantity which has both magnitude and direction is called a vector quantity, provided that two such quantities can be combined by vector addition.

A vector quantity can be represented in the plane by a directed line segment, whose length is proportional to the magnitude of the vector and whose direction is the direction of the vector.

Centroid of Triangle:

\[\mathbf{\overline{g}}=\frac{\mathbf{\overline{a}}+\mathbf{\overline{b}}+\mathbf{\overline{c}}}{3}\]

Centroid of Tetrahedron:

\[\overline{\mathbf{g}}=\frac{\overline{\mathbf{a}}+\overline{\mathbf{b}}+\overline{\mathbf{c}}+\overline{\mathbf{d}}}{4}\]

Incentre of Triangle:

\[\overline{\mathrm{h}}=\frac{\left|\overline{\mathrm{AB}}\right|\overline{\mathrm{c}}+\left|\overline{\mathrm{BC}}\right|\overline{\mathrm{a}}+\left|\overline{\mathrm{AC}}\right|\overline{\mathrm{b}}}{\left|\overline{\mathrm{AB}}\right|+\left|\overline{\mathrm{BC}}\right|+\left|\overline{\mathrm{AC}}\right|}\]

Orthocentre of Triangle:

\[\overline{\mathrm{p}}=\frac{\tan A\left(\overline{\mathrm{a}}\right)+\tan B\left(\overline{\mathrm{b}}\right)+\tan C\left(\overline{\mathrm{c}}\right)}{\tan A+\tan B+\tan C}\]

\[\mathbf{\overline{r}}=\mathbf{\frac{m\overline{b}+n\overline{a}}{m+n}}\]

\[\overline{\mathrm{r}}=\frac{\mathrm{m\overline{b}-n\overline{a}}}{\mathrm{m-n}}\]

If R (r̄) is the mid-point of the line segment joining the points A (ā) and B (b̄), then

\[\overline{\mathbf{r}}=\frac{\overline{\mathbf{a}}+\overline{\mathbf{b}}}{2}\]

\[\cos\theta=\frac{\mathbf{a}\cdot\mathbf{b}}{|\mathbf{a}||\mathbf{b}|}\]

Projection vector of \[\vec{a} on \vec{b} = \left(\frac{\vec{a} \cdot \vec{b}}{|\vec{b}|^2}\right) \vec{b}, \vec{b} \neq \vec{0}\]

Projection vector of \[\vec{b} on \vec{a} = \left(\frac{\vec{a} \cdot \vec{b}}{|\vec{a}|^2}\right) \vec{a}, \vec{a} \neq \vec{0}\]

cosθ \[= \frac{ \vec{a} \cdot \vec{b} }{ | \vec{a} | \, | \vec{b} | } = \frac{ \text{scalar product of the two vectors} }{ \text{product of their moduli} }\]

\[\vec{AB}=(x_2-x_1)\hat{i}+(y_2-y_1)\hat{j}\]

\[|\vec{AB}|=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}\]

\[\vec{a}\times(\vec{b}+\vec{c})=\vec{a}\times\vec{b}+\vec{a}\times\vec{c}\]

In 2D:

If \[\vec{a}=a_1\hat{i}+a_2\hat{j},\quad\vec{b}=b_1\hat{i}+b_2\hat{j}\]

\[\vec{a}\cdot\vec{b}=a_1b_1+a_2b_2\]

Angle Between Two Vectors (2D):

\[\cos\theta=\frac{a_1b_1+a_2b_2}{\left|\vec{a}\right|\left|\vec{b}\right|}\]

In 3D

If  \[\vec{a}=a_1\hat{i}+a_2\hat{j}+a_3\hat{k},\quad\vec{b}=b_1\hat{i}+b_2\hat{j}+b_3\hat{k}\]

\[\vec{a}\cdot\vec{b}=a_1b_1+a_2b_2+a_3b_3\]

1. Vector area of a triangle:

\[\text{Vector area of }\triangle ABC=\frac{1}{2}(\vec{AB}\times\vec{AC})\]

2. Collinearity condition

\[\vec{a}\times\vec{b}+\vec{b}\times\vec{c}+\vec{c}\times\vec{a}=\vec{0}\]

3. Area Parallelogram

\[\text{Area of parallelogram}=|\vec{a}\times\vec{b}|\]

If \[\mathrm{M}({\overline{m}})\] is the mid-point of the line segment joining the points \[\mathrm{A}({\overline{a}})\] and \[\mathrm{B}({\overline{b}})\] then \[\overline{m}=\frac{\left(\overline{a}+\overline{b}\right)}{2}\]

Scalar projection = \[\frac{\text{scalar product}}{\text{Modulus of vector}}\] 

\[\text{Scalar Projection of }\overline{b}\mathrm{~on~}\overline{a}=\frac{\overline{a}\cdot\overline{b}}{|\overline{a}|}\]

\[\text{Vector Projection of }\overline{b}\mathrm{~on~}\overline{a}=\left(\overline{a}\cdot\overline{b}\right)\frac{\overline{a}}{\left|\overline{a}\right|^{2}}\]

\[(a\hat{i}+b\hat{j})+(x\hat{i}+y\hat{j})=(a+x)\hat{i}+(b+y)\hat{j}\]

\[\lambda(a\hat{i}+b\hat{j})=\lambda a\hat{i}+\lambda b\hat{j}\]

\[\vec{OR}=\frac{m\vec{q}+n\vec{p}}{m+n}\]

\[\vec{OR}=\frac{m\vec{q}-n\vec{p}}{m-n}\]

If two vectors are represented by two sides of a triangle taken in order, then their sum is represented by the third side of the triangle taken in the same order.

If two vectors are represented by two adjacent sides of a parallelogram, then their resultant is represented by the diagonal passing through their common initial point.

The difference of two vectors is obtained by adding the negative of one vector.

Statement: 

If\[\vec{a},\vec{b},\vec{c},\] are three non-zero vectors and \[\vec{a}\times\vec{c}=\vec{b}\times\vec{c}\], then either \[\vec{a}=\vec{b}\mathrm{~or~}(\vec{a}-\vec{b})\] and \[\vec{c}\]are parallel vectors. 

\[\vec{a}\times\vec{c}=\vec{b}\times\vec{c}\Rightarrow\vec{a}=\vec{b}\mathrm{~or~}(\vec{a}-\vec{b})\parallel\vec{c}\]

The cross product has no cancellation law

If three points O, A, and B are so chosen that  \[\overrightarrow{OA}\] and  \[\overrightarrow{AB}\] respectively represent \[\overrightarrow{a}\] and \[\overrightarrow{b}\], then \[\overrightarrow{OB}\] is defined as the sum of  \[\overrightarrow{a}\] and \[\overrightarrow{b}\] and is written as \[\overrightarrow{c}=\overrightarrow{a}+\overrightarrow{b}\], where  \[\overrightarrow{c}\] stands for the vector \[\overrightarrow{OB}\]. \[\overrightarrow{c}\] or \[\overrightarrow{a}\] + \[\overrightarrow{b}\] is also called the resultant of  \[\overrightarrow{a}\] and \[\overrightarrow{b}\]. This is known as the Triangle law of vectors.

The result of adding two co-initial vectors is the vector represented by the diagonal of the parallelogram formed with the component vectors as adjacent sides. This is the Parallelogram Law of addition of vectors, which is thus a direct consequence of the triangle law.

Statement:

Two vectors in a plane are equal iff their x-components and y-components are equal.

If \[\vec{a}=a_1\hat{i}+a_2\hat{j},\quad\vec{b}=b_1\hat{i}+b_2\hat{j}\] Then \[\vec{a}=\vec{b}\] ⟺a_{1 ​}= b₁​ and a₂​ = b₂​

• Scalars have only magnitude.

• Vectors have magnitude and direction.

• Vectors are represented by directed line segments.

• \[\vec{AB}\] represents a vector from A to B.

• Magnitude of a vector is its length and is always non-negative.

• \[\vec{OP}\] is the position vector of point \[P(x, y, z)\].

• \[|\vec{OP}| = \sqrt{x^2 + y^2 + z^2}\].

• Direction angles are the angles a line makes with the positive coordinate axes.

• Direction cosines are \[\cos \alpha\], \[\cos \beta\], and \[\cos \gamma\].

• If direction cosines are (l, m, n), then \[l^2 + m^2 + n^2 = 1\].

• Direction ratios are any numbers proportional to direction cosines.

• If direction ratios are (a, b, c), then corresponding direction cosines are:

• For points \[A(x_1, y_1, z_1)\], \[B(x_2, y_2, z_2)\], direction ratios of AB are \[(x_2 - x_1, y_2 - y_1, z_2 - z_1)\].

• Angle between two lines can be found using either direction cosines or direction ratios.

• A vector has both magnitude and direction.

• Resultant means the combined effect of two or more vectors.

• Triangle law uses head-to-tail arrangement.

• Parallelogram law uses adjacent sides from the same initial point.

• Vector addition is commutative and associative.

• Zero vector is the identity element for vector addition.

• Difference of vectors is obtained by adding the negative of a vector.

Special Cases

• Perpendicular → \[\overline{\mathrm{a}}\cdot\overline{\mathrm{b}}=0\]

• Parallel → \[\mathbf{\overline{a}}\cdot\mathbf{\overline{b}}=\mathbf{ab}\]

Projection

• Scalar= \[\frac{\mathbf{a}\cdot\mathbf{b}}{|\mathbf{b}|}\]

• Vector \[=\frac{\mathbf{a}\cdot\mathbf{b}}{|\mathbf{b}|^{2}}\cdot\mathbf{b}\]

If \[\vec{a}\] and \[\vec{b}\] are two vectors,

\[\vec{a}-\vec{b}=\vec{a}+(-\vec{b})\]

Let\[\vec{a}\] be a vector and m a scalar. Then m\[\vec{a}\] is called the product of  \[\vec{a}\] by the scalar m.

Properties:

• The direction of m\[\overrightarrow{a}\] is the same as or parallel to that of \[\overrightarrow{a}\].

• The magnitude of m\[\overrightarrow{a}\] is given by

  \[|m\vec{a}|=|m||\vec{a}|\]

• The sense of m→a is:

  • same as \[\vec{a}\], if m is positive

  • opposite to \[\vec{a}\], if m is negative

If\[\vec{a}\] and \[\vec{b}\] are two non-zero vectors, then \[\vec{a}\cdot\vec{b}=|\vec{a}||\vec{b}|\cos\theta\]

Cases:

• Acute angle (0< θ < \[\frac{\pi}{2}\])
  cos⁡θ > 0  ⇒  \[\vec{a}\cdot\vec{b}>0\]
  

• Right angle (θ = \[\frac{\pi}{2}\]
  cos⁡θ = 0  ⇒  \[\vec{a}\cdot\vec{b}=0\]
  

• Obtuse angle (\[\frac{\pi}{2}\] < θ ≤ π)

  cos⁡θ < 0  ⇒  \[\vec{a}\cdot\vec{b}<0\]

• Position of dot & cross doesn’t matter
  \[(\vec{a}\times\vec{b})\cdot\vec{c}=\vec{a}\cdot(\vec{b}\times\vec{c})\]
  

• Cyclic order unchanged ⇒ STP unchanged
  \[[\vec{a}\operatorname{\vec{b}}\vec{c}]=[\vec{b}\operatorname{\vec{c}}\vec{a}]=[\vec{c}\operatorname{\vec{a}}\vec{b}]\]
  

• Interchanging two vectors changes the sign
  \[[\vec{a}\operatorname{\vec{b}}\vec{c}]=-\left[\vec{b}\operatorname{\vec{a}}\vec{c}\right]\]
  

• If any two vectors are equal
  \[[\vec{a}\operatorname{\vec{a}}\vec{b}]=0\]

• If any two vectors are parallel
  \[[\vec{a}\operatorname{\vec{b}}\operatorname{\vec{c}}]=0\]

If: \[\vec{a}=a_1\hat{i}+a_2\hat{j}+a_3\hat{k}\]

Then:

Addition: \[\vec{a}+\vec{b}=(a_1+b_1)\hat{i}+(a_2+b_2)\hat{j}+(a_3+b_3)\hat{k}\]

Scalar Multiplication: \[\lambda\vec{a}=\lambda a_1\hat{i}+\lambda a_2\hat{j}+\lambda a_3\hat{k}\]