Revision: Applications of Vector Algebra Mathematics HSC Science Class 12 Tamil Nadu Board of Secondary Education

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

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

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

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

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.

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

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

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

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

The scalar product or dot product of two nonzero vectors \[\vec P\] and \[\vec Q\] is defined as the product of the magnitudes of the two vectors and the cosine of the angle θ between the two vectors.

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

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

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

\[\vec P\] ⋅ \[\vec Q\] ​= PQ cos θ

∣\[\vec P\] × \[\vec Q\]​∣ = PQ sin θ

\[\vec{A}\cdot\vec{B}=|\vec{A}||\vec{B}|\cos\theta=AB\cos\theta\]

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.