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

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

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:

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.

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.

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 P(x, y, z) is a point, then its position vector is

This is called the component form of a vector.

If \[P_1(x_1, y_1, z_1)\] and \[P_2(x_2, y_2, z_2)\] are two points in space, then the vector joining \[P_1\] to \[P_2\] is the vector 

\[\vec{P_1P_2}\]

representing the displacement from \[P_1\] (initial point) to \[P_2\] (terminal point).

Magnitude of vector: 

Projection is the part of one vector in the direction of another vector.

Scalar projection of \[\vec{a}\] on \[\vec{b}\]

Vector projection of \[\vec{a}\] on \[\vec{b}\]

If \[\vec{a}\] and \[\vec{b}\] are two vectors and \[\theta\] is the angle between them, then their scalar product is given by:

If \[\vec{a}\] and \[\vec{b}\] are two vectors with angle \[\theta\] between them, then their vector product is:

where \[\hat{n}\] is a unit vector perpendicular to both \[\vec{a}\] and \[\vec{b}\], in the direction given by the right-hand rule.

Cross Product Angle: \[\sin \theta = \frac{|\vec{a} \times \vec{b}|}{|\vec{a}| |\vec{b}|}\]

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.

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.

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

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

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

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

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:

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

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

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

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

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

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

Parallelepiped: Volume = [a b c]

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

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.

• 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 \[P(x, y, z)\].

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

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:

• Initial point: starting point of vector; terminal point: ending point.

• Vector joining \[P_1(x_1, y_1, z_1)\] to \[P_2(x_2, y_2, z_2)\]:

  \[\vec{P_1P_2} = (x_2 - x_1)\hat{i} + (y_2 - y_1)\hat{j} + (z_2 - z_1)\hat{k}\]

• Order matters: \[\vec{P_1P_2} = -\vec{P_2P_1}\]

• Magnitude equals distance between points:

  \[|\vec{P_1P_2}| = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}\]

• Section formula gives the position vector of a point dividing a line segment in a given ratio.

• For internal division, use \(\dfrac{m\vec{b}+n\vec{a}}{m+n}\).

• For external division, use \(\dfrac{m\vec{b}-n\vec{a}}{m-n}\).

• Midpoint is the special case when the ratio is \(1:1\).

• Centroid formulas are natural extensions of the same averaging idea.

• Dot product result is a scalar.

• Cross product result is a vector.

• Dot product uses cosine; cross product uses sine.

• Dot product helps in angle and projection questions.

• Cross product helps in area and direction questions.

• \[\vec{a} \cdot \vec{b} = 0\] indicates perpendicular non-zero vectors.

• \[\vec{a} \times \vec{b} = \vec{0}\] indicates parallel vectors.

• Applications of Cross Product: 

  Area of Triangle:

  \[\frac{1}{2}|\vec{a} \times \vec{b}|\]

  Area of Parallelogram:

  \[|\vec{a} \times \vec{b}|\]

• 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.

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)