Vectors
Product of Two Vectors
- The angles between the two vectors \overrightarrow{a} = a_{1}\hat{i} + b_{1}\hat{j} + c_{1}\hat{k} and \overrightarrow{b} = a_{2}\hat{i} + b_{2}\hat{j} + c_{2}\hat{k} is \cos \theta = \frac{a_{1}a_{2} + b_{1}b_{2} + c_{1}c_{2}}{\sqrt{a_{1}^{2} + b_{1}^{2} + c_{1}^{2}}\sqrt{a_{2}^{2} + b_{2}^{2} + c_{2}^{2}}}
(or)
\cos \theta = \frac{\overrightarrow{a} \cdot \overrightarrow{b}}{|\overrightarrow{a}||\overrightarrow{b}|} - Two Vector \overrightarrow{a} and \overrightarrow{b} are parallel if \overrightarrow{a} = \lambda \overrightarrow{b}
(or)
\frac{a_{1}}{a_{2}} = \frac{b_{1}}{b_{2}} = \frac{c_{1}}{c_{2}}
if \overrightarrow{a} = a_{1}\hat{i} + b_{1}\hat{j} + c_{1}\hat{k}, \overrightarrow{b} = a_{2}\hat{i} + b_{2}\hat{j} + c_{2}\hat{k}
(or)
\overrightarrow{a} \times \overrightarrow{b} = 0 - Two Vectors \overrightarrow{a} = a_{1}\hat{i} + b_{1}\hat{j} + c_{1}\hat{k} \ {\tt and} \ \overrightarrow{b} = a_{2}\hat{i} + b_{2}\hat{j} + c_{2}\hat{k} are perpendicular if
\overrightarrow{a} \cdot \overrightarrow{b} = 0
(or)
a1a2 + b1b2 + c1c2 = 0 - Three vectors \overrightarrow{a} = a_{1}\hat{i} + b_{1}\hat{j} + c_{1}\hat{k}, \ \overrightarrow{b} = a_{2}\hat{i} + b_{2}\hat{j} + c_{2}\hat{k}, \ \overrightarrow{c} = a_{3}\hat{i} + b_{3}\hat{j} + c_{3}\hat{k} are coplanar if \left[\overrightarrow{a} \ \overrightarrow{b} \ \overrightarrow{c}\right] = 0
(or)
\begin{vmatrix}a_{1} & b_{1} & c_{1} \\ a_{2} & b_{2} & c_{2} \\ a_{3} & b_{3} & c_{3} \end{vmatrix} = 0 - Three Vectors \overrightarrow{a_{1}} = a_{1}\hat{i} + b_{1}\hat{j} + c_{1}\hat{k}, \ \overrightarrow{b_{1}} = a_{2}\hat{i} + b_{2}\hat{j} + c_{2}\hat{k}, {\tt and} \ \overrightarrow{c_{1}} = a_{3}\hat{i} + b_{3}\hat{j} + c_{3}\hat{k} are linear independent vectors if three exists scalars λ1, λ2, λ3 such that \lambda_{1}\overrightarrow{a_{1}} + \lambda_{2}\overrightarrow{b_{1}} + \lambda_{3}\overrightarrow{c_{1}} = 0 \Rightarrow \lambda_{1} = \lambda_{2} = \lambda_{3} = 0
(or)
\begin{vmatrix}a_{1} & b_{1} & c_{1} \\ a_{2} & b_{2} & c_{2} \\ a_{3} & b_{3} & c_{3} \end{vmatrix} \neq 0 - Three Vector \overrightarrow{a_{1}} = a_{1}\hat{i} + b_{1}\hat{j} + c_{1}\hat{k}, \ \overrightarrow{b_{1}} = a_{2}\hat{i} + b_{2}\hat{j} + c_{2}\hat{k}, {\tt and} \ \overrightarrow{c_{1}} = a_{3}\hat{i} + b_{3}\hat{j} + c_{3}\hat{k} are linearly independent vectors if there exists scalars λ1, λ2, λ3 such that \lambda_{1} \overrightarrow{a_{1}} + \lambda_{2} \overrightarrow{b_{1}} + \lambda_{3} \overrightarrow{c_{1}} = 0
atleast one of λi ≠ 0 (or)
\begin{vmatrix}a_{1} & b_{1} & c_{1} \\ a_{2} & b_{2} & c_{2} \\ a_{3} & b_{3} & c_{3} \end{vmatrix} = 0 - The unit vector perpendicular to the two vectors \overrightarrow{a} and \overrightarrow{b} is \pm \left(\frac{\overrightarrow{a} \times \overrightarrow{b}}{|\overrightarrow{a} \times \overrightarrow{b}|}\right)
- The unit vector perpendicular to the plane formed by three points A\left(\overrightarrow{a}\right), B\left(\overrightarrow{b}\right), C\left(\overrightarrow{c}\right) is = \pm \left(\frac{\overrightarrow{AB} \times \overrightarrow{BC}}{|\overrightarrow{AB} \times \overrightarrow{BC}|}\right)
- The position vector of the Incentre (I) of the triangle with vertices A (\overrightarrow{a}), B (\overrightarrow{b}), C (\overrightarrow{c}) is = \frac{a \cdot \overrightarrow{a} + b \cdot \overrightarrow{b} + c \cdot \overrightarrow{c}}{a + b + c}
where a = |\overrightarrow{BC}|, b = |\overrightarrow{CA}|, c = |\overrightarrow{AB}| - The position Vectors of Excentres (I1, I2, I3) of triangle with Vertices A(\overrightarrow{a}), B(\overrightarrow{b}), C(\overrightarrow{c}) is
\overrightarrow{OI_{1}} = \frac{-a \cdot \overrightarrow{a} + b \cdot \overrightarrow{b} + c \cdot \overrightarrow{c}}{-a + b + c}
\overrightarrow{OI_{2}} = \frac{a \cdot \overrightarrow{a} - b \cdot \overrightarrow{b} + c \cdot \overrightarrow{c}}{a - b + c}
\overrightarrow{OI_{3}} = \frac{a \cdot \overrightarrow{a} + b \cdot \overrightarrow{b} - c \cdot \overrightarrow{c}}{-a + b - c}
where a = |\overrightarrow{BC}|, b = |\overrightarrow{CA}|, c = |\overrightarrow{AB}| - If \overrightarrow{a} and \overrightarrow{b} are any two vectors and θ is angle between \overrightarrow{a} and \overrightarrow{b} then
\overrightarrow{a}\cdot \overrightarrow{b} > 0 \Rightarrow θ is acute
\overrightarrow{a}\cdot \overrightarrow{b} < 0 \Rightarrow θ is obtuse
\overrightarrow{a}\cdot \overrightarrow{b} = 0 \Rightarrow θ is right angled - The scalar projection (orthogonal projection) (component) of \overrightarrow{a} on \overrightarrow{b} is \frac{\overrightarrow{a} \cdot \overrightarrow{b}}{|\overrightarrow{b}|}
- The vector projection (orthogonal projection) of \overrightarrow{a} and \overrightarrow{b} is \frac{\left(\overrightarrow{a} \cdot \overrightarrow{b}\right)\overrightarrow{b}}{|\overrightarrow{b}|^{2}}
- The orthogonal projection of \overrightarrow{a} perpendicular to \overrightarrow{b} is = \overrightarrow{a} - \frac{(\overrightarrow{a} \cdot \overrightarrow{b}) \overrightarrow{b}}{|\overrightarrow{b}|}
- A constant force \overrightarrow{F} acting on a particle diplaces if from A to B, then work done is w = \overrightarrow{F} \cdot \overrightarrow{AB}
- If \overrightarrow{F} is the resultant of the forces \overrightarrow{F_{1}}, \overrightarrow{F_{2}}, \overrightarrow{F_{3}} ...... \overrightarrow{F_{n}} then the work done in displacing the particle form A to B is
w = \left(\overrightarrow{F_{1}}, \overrightarrow{F_{2}}+ ...... + \overrightarrow{F_{n}}\right)\cdot \overrightarrow{AB} - A vector which is perpendicular to both the vector \overrightarrow{a} and \overrightarrow{b} is \overrightarrow{a} \times \overrightarrow{b}
- If a vector \overrightarrow{r} is perpendicular to both \overrightarrow{a} and \overrightarrow{b} then \overrightarrow{r} = \lambda (\overrightarrow{a} \times \overrightarrow{b})
- The relation between the dot and cross product of two vectors \overrightarrow{a} and \overrightarrow{b} is |\overrightarrow{a} \times \overrightarrow{b}|^{2} = |\overrightarrow{a}|^{2} |\overrightarrow{b}|^{2} - (\overrightarrow{a} \cdot \overrightarrow{b})^{2}
- \overrightarrow{a}, \overrightarrow{b}, \overrightarrow{c} are three non zero vectors and \overrightarrow{a} \times \overrightarrow{c} = \overrightarrow{b} \times \overrightarrow{c} \Rightarrow \ {\tt either} \ \overrightarrow{a} = \overrightarrow{b}\ {\tt or} \ \overrightarrow{a} - \overrightarrow{b} || \overrightarrow{c}
- The area of the triangle formed by the vertices A(\overrightarrow{a}), B(\overrightarrow{b}), C(\overrightarrow{c}) is
= \frac{1}{2}|\overrightarrow{AB} \times \overrightarrow{BC}| - The area of the triangle whose adjacent sides are given by the vectors \overrightarrow{a} and \overrightarrow{b} is
= \frac{1}{2}|\overrightarrow{a} \times \overrightarrow{b}| - The area of the parallelogram formed by the vectors \overrightarrow{a} and \overrightarrow{b} as adjacent sides is
= |\overrightarrow{a} \times \overrightarrow{b}| - The area of the paralleogram formed by the vectors \overrightarrow{d_{1}} and \overrightarrow{d_{2}} as diagonals is
= \frac{1}{2}|\overrightarrow{d_{1}} \times \overrightarrow{d_{2}}| - Three points A(\overrightarrow{a}), B(\overrightarrow{b}), C(\overrightarrow{c}) are collinear if (\overrightarrow{a} \times \overrightarrow{b}) + (\overrightarrow{b} \times \overrightarrow{c}) + (\overrightarrow{c} \times \overrightarrow{a}) = \overrightarrow{o}
- For any vector \overrightarrow{a}
(\overrightarrow{a} \times \overrightarrow{i})^{2} + (\overrightarrow{a} \times \overrightarrow{j})^{2} + (\overrightarrow{a} \times \overrightarrow{k})^{2} = 2|\overrightarrow{a}|^{2} - If \overrightarrow{a}, \overrightarrow{b}, \overrightarrow{c} are three mutually perpendicular vectors, then \overrightarrow{a} \times \overrightarrow{b}, \overrightarrow{b} \times \overrightarrow{c}, \overrightarrow{c} \times \overrightarrow{a} are also mutually perpendicular vectors
- Let \overrightarrow{OP} = \overrightarrow{r} be the position vector of the point P on the line of action of force F. Then the moment of force F about 'O' is given by \overrightarrow{r} \times \overrightarrow{F}
- If \overrightarrow{a}, \overrightarrow{b}, \overrightarrow{c} are right handed system of vectors \Rightarrow \left[\overrightarrow{a} \ \overrightarrow{b} \ \overrightarrow{c}\right] > 0
- If \overrightarrow{a}, \overrightarrow{b}, \overrightarrow{c} are left handed system of vectors \Rightarrow \left[\overrightarrow{a} \ \overrightarrow{b} \ \overrightarrow{c}\right] < 0
- If \overrightarrow{a}, \overrightarrow{b}, \overrightarrow{c} are mutually perpendicular unit vectors, then \left[\overrightarrow{a} \ \overrightarrow{b} \ \overrightarrow{c}\right] = \pm 1
- The volume of a parallelepiped having coterminous edges \overrightarrow{a}, \overrightarrow{b}, \overrightarrow{c} is = \mid [\overrightarrow{a} \ \overrightarrow{b} \ \overrightarrow{c}]\mid
- The volume of a tetrahedron having coterminous adges \overrightarrow{a}, \overrightarrow{b}, \overrightarrow{c} is = \frac{1}{6} |[\overrightarrow{a} \ \overrightarrow{b} \ \overrightarrow{c}]|
- The volume of a parallelopiped having vertices A(\overrightarrow{a}), B(\overrightarrow{b}), C(\overrightarrow{c}), D(\overrightarrow{d}) is = |[\overrightarrow{AB} \ \overrightarrow{AC} \ \overrightarrow{AD}]|
- The volume of a Tetrahedron having vertices A(\overrightarrow{a}), B(\overrightarrow{b}), C(\overrightarrow{c}), D(\overrightarrow{d}) is = \frac{1}{6} |[\overrightarrow{AB} \ \overrightarrow{AC} \ \overrightarrow{AD}]|
- Four points A(\overrightarrow{a}), B(\overrightarrow{b}), C(\overrightarrow{c}), D(\overrightarrow{d}) are coplanar if \left[\overrightarrow{AB} \ \overrightarrow{AC} \ \overrightarrow{AD}\right] = 0
- \left[\hat{i} \ \hat{j} \ \hat{k}\right] = \left[\hat{j} \ \hat{k} \ \hat{i}\right] = \left[\hat{k} \ \hat{i} \ \hat{j}\right] = 1
\left[\hat{i} \ \hat{k} \ \hat{j}\right] = \left[\hat{k} \ \hat{j} \ \hat{i}\right] = \left[\hat{j} \ \hat{i} \ \hat{k}\right] = -1 - If \overrightarrow{a} + \overrightarrow{b} + \overrightarrow{c} = \overrightarrow{o} then \tt \overrightarrow{a}\times \overrightarrow{b}=\overrightarrow{b}\times \overrightarrow{c}=\overrightarrow{c}\times \overrightarrow{a} i.e. If \overrightarrow{a} + \overrightarrow{b} + \overrightarrow{c} = \overrightarrow{o} then, \overrightarrow{a}, \overrightarrow{b}, \overrightarrow{c} are coplanar
- \left[\overrightarrow{a} + \overrightarrow{b} \ \ \ \overrightarrow{b} + \overrightarrow{c} \ \ \ \overrightarrow{c} + \overrightarrow{a}\right] = 2 [\overrightarrow{a} \ \overrightarrow{b} \ \overrightarrow{c}]
- \left[\overrightarrow{a} - \overrightarrow{b} \ \ \ \overrightarrow{b} - \overrightarrow{c} \ \ \ \overrightarrow{c} - \overrightarrow{a}\right] = 0
- \left[\overrightarrow{a} \times \overrightarrow{b} \ \ \ \overrightarrow{b} \times \overrightarrow{c} \ \ \ \overrightarrow{c} \times \overrightarrow{a}\right] = \left[\overrightarrow{a} \ \overrightarrow{b} \ \overrightarrow{c}\right]^{2}
- If \overrightarrow{a}, \overrightarrow{b}, \overrightarrow{c} are any three vectors, then \overrightarrow{a} \times (\overrightarrow{b} \times \overrightarrow{c}) = (\overrightarrow{a} \cdot \overrightarrow{c})\overrightarrow{b} - (\overrightarrow{a} \cdot \overrightarrow{b}) \overrightarrow{c}
(\overrightarrow{a} \times \overrightarrow{b}) \times \overrightarrow{c}= (\overrightarrow{a} \cdot \overrightarrow{c})\overrightarrow{b} - (\overrightarrow{b} \cdot \overrightarrow{c}) \overrightarrow{a} - (\overrightarrow{a} \times \overrightarrow{b}) \cdot (\overrightarrow{c} \times \overrightarrow{d}) = \begin{vmatrix}\overrightarrow{a} \cdot \overrightarrow{c} & \overrightarrow{a} \cdot \overrightarrow{d} \\ \overrightarrow{b} \cdot \overrightarrow{c} & \overrightarrow{b} \cdot \overrightarrow{d} \end{vmatrix}
- (\overrightarrow{a} \ \overrightarrow{b} \ \overrightarrow{c}) \ (\overrightarrow{l} \ \overrightarrow{m} \ \overrightarrow{n}) = \begin{vmatrix}\overrightarrow{a} \cdot \overrightarrow{l} & \overrightarrow{b} \cdot \overrightarrow{l} & \overrightarrow{c} \cdot \overrightarrow{l} \\ \overrightarrow{a} \cdot \overrightarrow{m} & \overrightarrow{b} \cdot \overrightarrow{m}& \overrightarrow{c} \cdot \overrightarrow{m} \\ \overrightarrow{a} \cdot \overrightarrow{n} & \overrightarrow{b} \cdot \overrightarrow{n}& \overrightarrow{c} \cdot \overrightarrow{n} \end{vmatrix}
- (\overrightarrow{a} \times \overrightarrow{b}) \times (\overrightarrow{c} \times \overrightarrow{d}) = [\overrightarrow{a} \ \overrightarrow{c} \ \overrightarrow{d}]\overrightarrow{b} - [\overrightarrow{b} \ \overrightarrow{c} \ \overrightarrow{d}]\overrightarrow{a}
= [\overrightarrow{a} \ \overrightarrow{b} \ \overrightarrow{d}]\overrightarrow{c} - [\overrightarrow{a} \ \overrightarrow{b} \ \overrightarrow{c}]\overrightarrow{d} - If \overrightarrow{a}, \overrightarrow{b}, \overrightarrow{c} are any three vectors, then \overrightarrow{a} \times \left(\overrightarrow{b} \times \overrightarrow{c}\right) + \overrightarrow{b} \times \left(\overrightarrow{c} \times \overrightarrow{a}\right) + \overrightarrow{c} \times \left(\overrightarrow{a} \times \overrightarrow{b}\right) = 0
- \overrightarrow{a} \times \left(\overrightarrow{b} \times \overrightarrow{c}\right) is a vector lies in the plane of \overrightarrow{b} and \overrightarrow{c} (or) parallel to the plane of \overrightarrow{b} and \overrightarrow{c}
- If \overrightarrow{a}, \overrightarrow{b}, \overrightarrow{c} are three non coplanar vectors and \overrightarrow{a}^{1}, \overrightarrow{b}^{1}, \overrightarrow{c}^{1} are reciprocul vectors of \overrightarrow{a}, \overrightarrow{b}, \overrightarrow{c} respectively.
\overrightarrow{a}^{1} = \frac{\overrightarrow{b} \times \overrightarrow{c}}{[\overrightarrow{a} \ \overrightarrow{b} \ \overrightarrow{c}]}
\overrightarrow{b}^{1} = \frac{\overrightarrow{c} \times \overrightarrow{a}}{[\overrightarrow{a} \ \overrightarrow{b} \ \overrightarrow{c}]}
\overrightarrow{c}^{1} = \frac{\overrightarrow{a} \times \overrightarrow{b}}{[\overrightarrow{a} \ \overrightarrow{b} \ \overrightarrow{c}]} - If \overrightarrow{a}^{1}, \overrightarrow{b}^{1}, \overrightarrow{c}^{1} are reciprocal vectors of \overrightarrow{a}, \overrightarrow{b}, \overrightarrow{c} then
\overrightarrow{a} \cdot \overrightarrow{a}^{1} = \overrightarrow{b} \cdot \overrightarrow{b}^{1} = \overrightarrow{c} \cdot \overrightarrow{c}^{1} = 1
\overrightarrow{a} \cdot \overrightarrow{b}^{1} = \overrightarrow{b} \cdot \overrightarrow{c}^{1} = \overrightarrow{c} \cdot \overrightarrow{a}^{1} = 0
Part1: View the Topic in this video From 02:00 To 54:06
Part2: View the Topic in this video From 01:40 To 53:53
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1. (a + b) · (a − b) = |a|2 − |b|2
2. |a + b|2 = |a|2 + |b|2 + 2 (a · b)
3. |a − b|2 = |a|2 + |b|2 − 2 (a · b)
4. |a + b|2 + |a − b|2 = (|a|2 + |b|2) and |a + b|2 + |a − b|2 = 4 (a · b)
5. If |a + b| = |a| + |b|, then a is parallel to b.
6. If |a + b| = |a − b|, then a is parallel to b.
7. (a · b)2 ≤ |a|2 |b|2
8. If θ is angle between two non-zero vectors, a, b, then we have
a · b = |a| |b| cos θ
\cos \theta = \tt \frac{a\cdot b}{|a| |b|}
9. If a = a1i + a2j + a3k and b = b1i + b2j + b3k. Then, the angle θ between a and b is given by \cos \theta = \tt \frac{a\cdot b}{|a| |b|}=\frac{a_{1}b_{1}+a_{2}b_{2}+a_{3}b_{3}}{\sqrt{a_1^2+a_2^2+a_3^2} \sqrt{b_1^2+b_2^2+b_3^2}}
10. Projection of a on b = \tt \frac{a\cdot b}{|a|}
11. Projection of b on a = \tt \frac{a\cdot b}{|a|}
12. Vector component of a vector a on b
=\tt \frac{a\cdot b}{|b|}\cdot\hat{b}=\frac{a\cdot b}{|b|}\cdot\frac{b}{|b|}=\frac{(a\cdot b)}{|b^{2}|} b
Similarly, the vector component of b on a =\tt \frac{(a\cdot b)}{|a^{2}|} a
13. If θ is the angle between two vectors \vec{a} and \vec{b}, then their cross product is given as \vec{a}\times \vec{b}=|\vec{a}||\vec{b}|\sin \theta \ \hat{n}