Definitions [8]
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:
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 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
\[\overset{-}{\operatorname*{\operatorname*{a}}}\times\left(\overset{-}{\operatorname*{\operatorname*{b}}}\times\overset{-}{\operatorname*{\operatorname*{c}}}\right)=\left(\overset{-}{\operatorname*{\operatorname*{a}}}\cdot\overset{-}{\operatorname*{\operatorname*{c}}}\right)\overline{b}-\left(\overset{-}{\operatorname*{\operatorname*{a}}}\cdot\overline{b}\right)\overline{c}\]
Formulae [2]
\[\vec{A}\cdot\vec{B}=|\vec{A}||\vec{B}|\cos\theta=AB\cos\theta\]
Parallelepiped: Volume = [a b c]
Tetrahedron: \[\frac{1}{6}\] [a b c]
Theorems and Laws [2]
Using properties of scalar triple product, prove that `[(bara + barb, barb + barc, barc + bara)] = 2[(bara, barb, barc)]`.
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
Prove by vector method, that the angle subtended on semicircle is a right angle.
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" = bar"CB" = 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.
Key Points
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Scalars have only magnitude.
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Vectors have magnitude and direction.
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Vectors are represented by directed line segments.
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\[\vec{AB}\] represents a vector from A to B.
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Magnitude of a vector is its length and is always non-negative.
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\[\vec{OP}\] is the position vector of point \[P(x, y, z)\].
-
\[|\vec{OP}| = \sqrt{x^2 + y^2 + z^2}\].
| Case | Vector Form | Cartesian Form |
|---|---|---|
| 1. Normal form (given normal vector) | \[\overline{\mathbf{r}}.\hat{\mathbf{n}}=\mathbf{p}\] | ax + by + cz + d = 0 |
| 2. Through a point (x₁, y₁, z₁) | \[\begin{bmatrix} \mathbf{\overline{r}}-\mathbf{\overline{a}} \end{bmatrix}.\mathbf{\overline{n}}=\mathbf{0}\] | a(x−x₁) + b(y−y₁) + c(z−z₁) = 0 |
| 3. Through point + parallel to two vectors | \[\begin{bmatrix} \overline{\mathbf{r}}\overline{\mathbf{b}}\overline{\mathbf{c}} \end{bmatrix}= \begin{bmatrix} \overline{\mathbf{a}}\overline{\mathbf{b}}\overline{\mathbf{c}} \end{bmatrix}\] | \[\begin{vmatrix} x-x_1 & y-y_1 & z-z_1 \\ \mathbf{b}_1 & \mathbf{b}_2 & \mathbf{b}_3 \\ \mathbf{c}_1 & \mathbf{c}_2 & \mathbf{c}_3 \end{vmatrix}=0\] |
| 4. Through three non-collinear points | \[(\mathbf{r-a})\cdot[(\mathbf{b-a})\times(\mathbf{c-a})]=0\] | \[\begin{vmatrix} x-x_1 & y-y_1 & z-z_1 \\ x_2-x_1 & y_2-y_1 & z_2-z_1 \\ x_3-x_1 & y_3-y_1 & z_3-z_1 \end{vmatrix}=0\] |
| 5. Through the intersection of two planes | \[\left(\overline{\mathbf{r}}.\overline{\mathbf{n}}_1-\mathbf{d}_1\right)+\lambda\left(\overline{\mathbf{r}}.\overline{\mathbf{n}}_2-\mathbf{d}_2\right)=0\] | (a₁x + b₁y + c₁z + d₁) + λ(a₂x + b₂y + c₂z + d₂) = 0 |
Equation of a Plane in Intercept form:
\[\frac{x}{a}+\frac{y}{b}+\frac{z}{c}=1\]
Distance of the Plane from Origin is
\[d=\frac{1}{\sqrt{\frac{1}{a^{2}}+\frac{1}{b^{2}}+\frac{1}{c^{2}}}}\]
Concepts [10]
- Basic Concepts of Vector Algebra
- Geometric Introduction to Vectors
- Multiplication of Vectors>Scalar Product(Dot Product)
- Scalar Triple Product
- Vector Triple Product
- Jacobi’S Identity and Lagrange’S Identity
- Application of Vectors to 3-dimensional Geometry
- Different Forms of Equation of a Plane
- Image of a Point in a Plane
- Meeting Point of a Line and a Plane
