Revision: Vectors Maths and Stats HSC Science (General) 12th Standard Board Exam Maharashtra State Board

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 \[[\vec{a}\vec{b}\vec{c}]\mathrm{~or~}[a,b,c]\]

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. 

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

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

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.

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:

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: 

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

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:

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

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

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:

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

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

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

Parallelepiped: Volume = [a b c]

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

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.

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.

• 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
  \[[\vec{a}\operatorname{\vec{b}}\operatorname{\vec{c}}]=0\]

• 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 \[P(x, y, z)\].

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

• 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.

• 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.