Revision: 12th Std >> Vectors MAH-MHT CET (PCM/PCB) Vectors

Scalar: Those quantities which have magnitude but no direction are called scalar quantities or scalars. e.g. length, mass, volume, temperature, work, etc.

Vector: Those quantities which have magnitude as well as direction are called vector quantities or vectors. e.g. force, displacement, velocity, etc.

Representation A vector is represented as a or bold letter (a).

A vector that has a magnitude of one unit and is used to indicate direction is called a unit vector.

A physical quantity that is described with both magnitude and direction is called a vector.

Vectors that are perpendicular to each other are called orthogonal vectors.

Vectors that act in the same plane are called coplanar vectors.

When a vector \[\vec P\] is split into two mutually perpendicular parts along the horizontal and vertical axes, those parts are called rectangular components.

A physical quantity that is described with magnitude alone is called a scalar.

A vector that has the same magnitude as a given vector but acts in the opposite direction is called a negative vector.

A vector whose magnitude is zero is called a zero vector.

A vector that describes the position of a point with respect to the origin is called a position vector.

Two vectors having the same magnitude and the same direction are called equal vectors.

In general, if a₁, a₂, …, aₙ are n vectors and t₁, t₂, …, tₙ are n scalars, then linear combination of vectors a₁, a₂, …, aₙ is t₁a₁ + t₂a₂ + … + tₙaₙ.

• For 2 vectors:

  \[\overline{\mathbf{r}}=x\overline{\mathbf{a}}+y\overline{\mathbf{b}}\]

• For 3 vectors:

  \[\mathbf{\overline{r}}=x\mathbf{\overline{a}}+y\mathbf{\overline{b}}+\mathbf{z}\mathbf{\overline{c}}\]

Two vectors a and b are collinear if there exists a scalar λ such that a = λb.

Three points A(a), B(b) and C(c) are collinear iff ∃ non-zero scalars x, y, z such that xa + yb + zc = 0, where x + y + z = 0.

Three points A(a), B(b) and C(c) are collinear if AB × BC = 0 i.e. a × b + b × c + c × a = 0.

a and b are two non-collinear vectors. A vector r is coplanar with a and b if and only if there exists a unique scalar λ₁ and λ₂ such that r = λ₁a + λ₂b

Three vectors a₁i + a₂j + a₃k, b₁i + b₂j + b₃k and c₁i + c₂j + c₃k are coplanar, if \[\begin{vmatrix} a_{1} & a_{2} & a_{3} \\ b_{1} & b_{2} & b_{3} \\ c_{1} & c_{2} & c_{3} \end{vmatrix}=0.\]

Four points with position vectors a = a₁i + a₂j + a₃k, b = b₁i + b₂j + b₃k, c = c₁i + c₂j + c₃k and d = d₁i + d₂j + d₃k will be coplanar iff

\[\begin{vmatrix} a_1 & a_2 & a_3 & 1 \\ b_1 & b_2 & b_3 & 1 \\ c_1 & c_2 & c_3 & 1 \\ d_1 & d_2 & d_3 & 1 \end{vmatrix}=0.\]

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.

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: \[\cos\alpha=\frac{x}{r},\quad\cos\beta=\frac{y}{r},\quad\cos\gamma=\frac{z}{r}\]

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 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 vector product of two non-null and non-parallel vectors a and b is expressed as:

a × b = |a||b| sinθ n̂ = ab sinθ n̂

The unit vector n̂ along a × b is given by:

\[\hat{\mathbf{n}}=\frac{\mathbf{a}\times\mathbf{b}}{|\mathbf{a}\times\mathbf{b}|}\]

The scalar triple product of three vectors a, b, and c is defined as

(a × b) · c = |a| |b| |c| sinθ cosφ,

where θ is the angle between a and b, and φ is the angle between a × b and c. It is also defined as [a b c].

For vectors \[\overline{a}\], \[\overline{b}\] and \[\overline{c}\] in the space, we define the vector triple product as

\[\overset{-}{\operatorname*{\operatorname*{a}}}\times\left(\overset{-}{\operatorname*{\operatorname*{b}}}\times\overset{-}{\operatorname*{\operatorname*{c}}}\right)=\left(\overset{-}{\operatorname*{\operatorname*{a}}}\cdot\overset{-}{\operatorname*{\operatorname*{c}}}\right)\overline{b}-\left(\overset{-}{\operatorname*{\operatorname*{a}}}\cdot\overline{b}\right)\overline{c}\]

\[\vec P\] ⋅ \[\vec Q\] ​= PQ cos θ

∣\[\vec P\] × \[\vec Q\]​∣ = PQ sin θ

If two vectors \[\vec P\] and \[\vec Q\]​ act at an angle θ, the magnitude of their resultant is:

If a vector \[\vec P\] is resolved into two rectangular components:

• Horizontal component: P_x = P cos⁡ θ
• Vertical component: P_y = P sin ⁡θ

Magnitude of Vector: \[\mid r\mid=\sqrt{x^{2}+y^{2}}\]

\[\theta=\tan^{-1}\left(\frac{y}{x}\right)\]

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

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

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

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

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:

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

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

\[\sin\theta=\frac{\left|\overline{a}\times\overline{b}\right|}{\left|\overline{a}\right|\left|\overline{b}\right|}\]

Parallelepiped: Volume = [a b c]

Tetrahedron: \[\frac{1}{6}\] [a b c]

The resultant R of two vectors \[\vec P\] and \[\vec Q\]​ always lies between their difference and their sum:

For any two vectors \[\vec P\] and \[\vec Q\]​:

The commutative law holds true for addition of vectors but not for subtraction.

For three vectors \[\vec P\], \[\vec Q\]​, and \[\vec R\]:

The associative law holds true for addition of vectors but not for subtraction.

If two vectors are represented in magnitude and direction by two sides of a triangle taken in order, then their sum is \[\overline{\mathrm{AC}}=\overline{\mathrm{AB}}+\overline{\mathrm{BC}}\]

If two co-initial vectors are represented in magnitude and direction by adjacent sides of a parallelogram, then their sum is represented in magnitude and direction by the diagonal of the parallelogram passing through the common point.

∴ \[\overline{\mathrm{AC}}=\overline{\mathrm{AB}}+\overline{\mathrm{BC}}\]

Two vectors can be added using either the Triangle Law or the Parallelogram Law. When two vectors \[\vec P\] and \[\vec Q\]​ are represented as two sides of a triangle (or two adjacent sides of a parallelogram), their resultant is represented by the third side (or the diagonal).

L.H.S = `[(bara + barb,  barb + barc,  barc + bara)]`

= `(bara + barb) . [(barb + barc) xx (barc + bara)]`

= `(bara + barb) . [barb xx barc + barb xx bara + barc xx barc + barc xx bara]`

= `(bara + barb) . [barb xx barc + barb xx bara + barc xx bara]   ...[∵ barc xx barc = bar0]`

= `bara . [(barb xx barc) + (barb xx bara) + (barc xx bara)] + barb . [(barb xx barc) + (barb xx bara) + (barc xx bara)]`

= `bara . (barb xx barc) + bara . (barb xx bara) + bara . (barc xx bara) + barb . (barb xx barc) + barb(barb xx bara) + barb(barc xx bara)`

= `[bara  barb  barc] + [bara  barb  bara] + [bara  barc  bara] + [barb  barb  barc] + [barb  barb  bara] + [barb  barc  bara]`

= `[bara  barb  barc] + 0 + 0 + 0 + 0 + [bara  barb  barc]`

= `2[bara  barb  barc]`

= R.H.S

Let seg AB be a diameter of a circle with centre C and P be any point on the circle other than A and B.

Then ∠APB is an angle subtended on a semicircle.

Let `bar"AC" = <sup>bar"CB"</sup> = bar"a"` and `bar"CP" = bar"r"`

Then `|bar"a"| = |bar"r"|`       ....(1)

`bar"AP" = bar"AC" + bar"CP"`

= `bar"a" + bar"r"`

= `bar"r" + bar"a"`

`bar"BP" = bar"BC" + bar"CP"`

= `- bar"CB" + bar"CP"`

= `- bar"a" + bar"r"`

∴ `bar"AP".bar"BP" = (bar"r" + bar"a").(bar"r" - bar"a")`

= `bar"r".bar"r" - bar"r".bar"a" + bar"a".bar"r" - bar"a".bar"a"`

= `|bar"r"|^2 - |bar"a"|^2`

= 0    ....`(∵ bar"r".bar"a" = bar"a".bar"r")`

∴ `bar"AP" ⊥ bar"BP"`

∴ ∠APB is a right angle.

Hence, the angle subtended on a semicircle is the right angle.

Consider the circle with the centre at O and AB is the diameter.

Let `bar(OA) = bar a, bar(OB) = bar b, bar(OC) = bar c`

∴ `|bar a| =|bar b| = |bar c| = r`    ...(1)

and `bar a = -bar b`    ...(2)

Consider:

`bar (AC) * bar (BC) = (bar c - bar a) * (bar c - bar b)`

= `(bar c - bar a) * (bar c + bar a)`    ...[From (2)]

= `|bar c|^2 - |bar a|^2`

= r² − r²    ...[From (1)]

= 0

∴ `bar(AC) * bar(BC) = 0`

∴ `bar(AC)` is perpendicular to `bar(BC)`

∴ ∠ACB = 90°

∴ Angle subtended on semi-circle is a right angle.

Coordinates on Axes

• X-axis → (x,0,0)(x, 0, 0)(x,0,0)

• Y-axis → (0,y,0)(0, y, 0)(0,y,0)

• Z-axis → (0,0,z)(0, 0, z)(0,0,z)

Coordinates on Planes

• XY-plane → (x,y,0)(x, y, 0)(x,y,0)

• YZ-plane → (0,y,z)(0, y, z)(0,y,z)

• ZX-plane → (x,0,z)(x, 0, z)(x,0,z)

Distance from Coordinate Planes

• From XY-plane → ∣z∣

• From YZ-plane → ∣x∣

• From ZX-plane → ∣y∣

Distance from Origin

\[\sqrt{x^{2}+y^{2}+z^{2}}\]

Distance Between Two Points

\[d=\sqrt{(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}+(z_{2}-z_{1})^{2}}\]

Octants:

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

1. Conversion: From D.R → D.C: \[l=\frac{a}{\sqrt{a^2+b^2+c^2}},m=\frac{b}{\sqrt{a^2+b^2+c^2}},n=\frac{c}{\sqrt{a^2+b^2+c^2}}\]

2. Angle between two lines

If direction cosines: \[\cos\theta=l_1l_2+m_1m_2+n_1n_2\]

If direction ratios: \[\cos\theta=\frac{a_1a_2+b_1b_2+c_1c_2}{\sqrt{a_1^2+b_1^2+c_1^2}\sqrt{a_2^2+b_2^2+c_2^2}}\]

3. If A(x₁, y₁, z₁), B(x₂, y₂, z₂):

\[\mathrm{D.Rs}=(x_2-x_1,y_2-y_1,z_2-z_1)\]

4. \[l^2+m^2+n^2=1\]

1. Determinant form:

If \[\overline{\mathrm{a}}=\mathrm{a}_{1}\hat{\mathrm{i}}+\mathrm{a}_{2}\hat{\mathrm{j}}+\mathrm{a}_{3}\hat{\mathrm{k}}\] and \[\overline{\mathrm{b}}=\mathrm{b}_1\hat{\mathrm{i}}+\mathrm{b}_2\hat{\mathrm{j}}+\mathrm{b}_3\hat{\mathrm{k}}\], then

\[\overline{\mathrm{a}}\times\overline{\mathrm{b}}= \begin{vmatrix} \hat{\mathrm{i}} & \hat{\mathrm{j}} & \hat{\mathrm{k}} \\ \mathbf{a}_1 & \mathbf{a}_2 & \mathbf{a}_3 \\ \mathbf{b}_1 & \mathbf{b}_2 & \mathbf{b}_3 \end{vmatrix}\]

2. Condition for zero cross product:

a × b = 0 ⇒ vectors are parallel (or one is zero)