Some of the most important formulas for vectors such as Scalar or Dot product of two vectors, vector or cross product of two vectors, etc. are provided here. The main objective of presenting the Product of Vectors formulas list is to simplify the vector multiplication problems at a faster pace. Therefore, get the formula sheet of Vector Product concept from below and compute them easily during homework, assignments, and exam preparation.

1. Dot Product

Product of two vectors is done by two methods when the product of two vectors results in a scalar quantity then it is called scalar product. It is also called as dot product because this product is represented by putting a dot (.).

2. Vector Product

When the product of two vectors results in a vector quantity then this product is called Vector Product. This product is represented by (×) sign so that it is also called as Cross Product.

3. Scalar or dot product of two vectors

a.b = |a| |b| cos θ = ab cos θProjection of b on a = \(\frac{a \cdot b}{|a|}\); Similarly projection of a on b = \(\frac{a \cdot b}{|b|}\)

4. Properties of scalar product

5. Angle between two vectors

(i). cos θ = \(\frac{a \cdot b}{|a||b|}=\hat{a}. \hat{b}\)

(iil) If a = a1i + a2j + a3k and b = b1i + b2j + b3k thencos θ = \(\frac{a_{1} b_{1}+a_{2} b_{2}+a_{3} b_{3}}{\sqrt{a_{1}^{2}+a_{2}^{2}+a_{3}^{2}} \sqrt{b_{1}^{2}+b_{2}^{2}+b_{3}^{2}}}\)

6. Components of b along & perpendicular to a

7. Work done by the force

If a constant force F acting on a particle displaces it from point A to B, then work done by the force W = f.d (where d = \(\overrightarrow{\mathrm{AB}}\))

8. Vector or cross product of two vectors

a × b = |a| |b| sin θ = ab sinθ

9. Vector product in terms of components

If a = a1i + a2j + a3k and b = b1i + b2j + b3k thena x b = (a2b3 – a3b2) i + (a3b1 – a1b3) j + (a1b2 – a2b1) k = \(\left|\begin{array}{lll}i & j & k \\a_{1} & a_{2} & a_{3} \\b_{1} & b_{2} & b_{3}\end{array}\right|\)

10. Angle between two vectors

If θ is the angle between a and b, then sin θ = \(\frac{|\mathbf{a} \times \mathbf{b}|}{|\mathbf{a}||\mathbf{b}|}\)If \(\hat{\mathrm{n}}\) is the unit vector perpendicular to the plane of a and b, then \(\hat{\mathrm{n}}\) = \(\frac{\mathbf{a} \times \mathbf{b}}{|\mathbf{a} \times \mathbf{b}|}\)Note:If i, j, k be three mutually perpendicular unit vectors, then

11. Properties of vector product

If a, b, c are any vectors and m, n any scalars then

12. Area of Triangle

13. Area of Parallelogram

14. Moment of Force

The moment of the force F acting at a point A about O is given by Moment of F = \(\overrightarrow{\mathrm{OA}} \times \overrightarrow{\mathrm{F}}=\overrightarrow{\mathrm{r}} \times \overrightarrow{\mathrm{F}}\)

15. Formula for scalar triple product

(i) If a = a1l + a2m + a3n, b = b1l + b2m + b3n and c = c1l + c2m + c3n, then[a b c] = \(\left|\begin{array}{lll}a_{1} & a_{2} & a_{3} \\b_{1} & b_{2} & b_{3} \\c_{1} & c_{2} & c_{3}\end{array}\right|\) = [ l m n]

(ii) For any three vectors a, b and c

16. Properties of scalar triple product

(i) The position of (.) and (×) can be interchangedi.e. a. (b × c) = (a × b). c but (a × b). c = c. (a × b)So [a b c] = [b c a] = [c a b]Therefore if we don’t change the cyclic order of a, b and c then the value of scalar triple product is not changed by interchanging dot and cross.

(ii) If the cyclic order of vectors is changed, then sign of scalar triple product is changed i.e.a. (b × c) = – a. (c × b) or [a b c] = – [a c b]from (i) and (ii) we have[abc] = [bca] = [cab] = -[acb]= -[bac]= -[cba]

(iii) The scalar triple product of three vectors when two of them are equal or parallel, is zero i.e.[a b b] = [a b a] = 0

(iv) The scalar triple product of three mutually perpendicular unit vectors is ±1.Thus [i j k] = 1, [i k j] = -1

(v) If two of the three vectors a, b, c are parallel then [a b c] = 0

(vi) a, b, c are three coplanar vectors if [a b c] = 0 i.e. the necessary and sufficient condition for three non-zero collinear vectors to be coplanar is [a b c] = 0

(vii) For any vectors a, b, c, d[a + b c d ] = [a c d] + [b c d]

17. Volume of Parallelopiped

If coterminous edges of a parallelopiped are a, b and c then volume = [a b c]

18. Volume of tetrahedron

(i) If a, b, c are position vectors of vertices A, B and C with respect to O, then volume of tetrahedron OABC = \(\frac{1}{6}\) [a b c]

(ii) If a, b, c, d are position vectors of vertices A, B, C, D of a tetrahedron ABCD, then its volume\(\left\{\begin{array}{ccc}\frac{1}{6}[\mathrm{AB} & \overrightarrow{\mathrm{AC}} & \overrightarrow{\mathrm{AD}}] \\& \mathrm{or} & \\\frac{1}{6}[\mathbf{b}-\mathrm{a} & \mathrm{c}-\mathrm{a} & \mathrm{d}-\mathrm{a}]\end{array}\right.\)

19. Vector triple product

(i) Definition:The vector triple product of three vectors a, b, c is defined as the vector product of two vectors a and b × c. It is denoted by a × (b × c).

(ii) Properties: Expansion formula for vector triple product is given bya × (b × c) = (a.c) b – (a.b) c(b × c) × a = (b.a) c – (c.a) b

20. (i) The vector ⊥ to both \(\overrightarrow{\mathrm{a}} \text { and } \overrightarrow{\mathrm{b}} \text { is } \overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}}\)

(ii) The unit vector ⊥ to the plane of \(\overrightarrow{\mathrm{a}} \text { and } \overrightarrow{\mathrm{b}}\) is = \(\frac{\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}}}{|\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}}|}\), and any vector of magnitude k in this direction is \(\frac{k(\vec{a} \times \vec{b})}{|\vec{a} \times \vec{b}|}\) [may be called a vector in the plane \(\overrightarrow{\mathbf{a}}, \overrightarrow{\mathbf{b}}\)]

(iii) If \(\overrightarrow { a, } \overrightarrow { b, } \overrightarrow { c } and\quad \overrightarrow { d } \) are coplanar then \([\vec{a} \vec{b} \vec{c}]=[\vec{d} \vec{a} \vec{b}]+[\vec{d} \vec{b} \vec{c}]+[\vec{d} \vec{c} \vec{a}]\)

(iv) \([\vec{a} \vec{b} \vec{c}][\vec{u} \vec{v} \vec{w}]\) = \(\left|\begin{array}{ccc}\vec{a} \cdot \vec{u} & \vec{b} \cdot \vec{u} & \vec{c} \cdot \vec{u} \\\vec{a} \cdot \vec{v} & \vec{b} \cdot \vec{v} & \vec{c} \cdot \vec{v} \\\vec{a} \cdot \vec{w} & \vec{b} \cdot \vec{w} & \vec{c} \cdot \vec{w}\end{array}\right|\)

(v) Scalar product of four vector’s: \((\vec{a} \times \vec{b}) \cdot(\vec{c} \times \vec{d})\) = \(\left|\begin{array}{cc}\vec{a} \cdot \vec{c} & \vec{b} \cdot \vec{c} \\\vec{a} \cdot \vec{d} & \vec{b} \cdot \vec{d}\end{array}\right|\)

(vi) Vector product of four vector’s then\((\vec{a} \times \vec{b}) \times(\vec{c} \times \vec{d})=[\vec{c} \vec{b} \vec{d}] \vec{c}-[\vec{a} \vec{b} \vec{c}] \vec{d}=[\vec{a} \vec{c} \vec{d}] \vec{b}-[\vec{b} \vec{c} \vec{d}] \vec{a}\)

(vii) An expression for any \(\overrightarrow{\mathrm{r}}\), in space, as a linear combination of three non coplanar vectors \(\overrightarrow{\mathrm{a}}, \overrightarrow{\mathrm{b}}, \overrightarrow{\mathrm{c}}\) then \(\vec{r}=\frac{[\vec{r} \vec{b} \vec{c}] \vec{a}+[\vec{r} \vec{c} \vec{a}] \vec{b}+[\vec{r} \vec{a} \vec{b}] \vec{c}}{[\vec{a} \vec{b} \vec{c}]}\)

(viii) Reciprocal system of vectors.If \(\overrightarrow{\mathbf{a}}, \overrightarrow{\mathbf{b}}, \overrightarrow{\mathbf{c}}\) be any three non coplanar vectors so that \([\vec{a} \vec{b} \vec{c}]\) ≠ 0 then the three vectors \(\overrightarrow{\mathbf{a}}^{\prime}, \overrightarrow{\mathbf{b}}^{\prime}, \overrightarrow{\mathbf{c}}^{\prime}\) define equation\(\overrightarrow{\mathrm{a}}^{\prime}=\frac{\overrightarrow{\mathrm{b}} \times \overrightarrow{\mathrm{c}}}{[\overrightarrow{\mathrm{a}} \overrightarrow{\mathrm{b}} \overrightarrow{\mathrm{c}}]}\), \(\overrightarrow{\mathrm{b}}^{\prime}=\frac{\overrightarrow{\mathrm{c}} \times \overrightarrow{\mathrm{a}}}{[\overrightarrow{\mathrm{a}} \overrightarrow{\mathrm{b}} \overrightarrow{\mathrm{c}}]}\), \(\overrightarrow{\mathrm{c}}^{\prime}=\frac{\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}}}{[\overrightarrow{\mathrm{a}} \overrightarrow{\mathrm{b}} \overrightarrow{\mathrm{c}}]}\) are called reciprocal system of vectors to the \(\overrightarrow{\mathrm{a}}, \overrightarrow{\mathrm{b}}, \overrightarrow{\mathrm{c}}\)

(ix) Properties of reciprocal system of vectors(a) \(\overrightarrow{\mathrm{a}} \cdot \overrightarrow{\mathrm{a}}^{\prime}=\overrightarrow{\mathrm{b}} \cdot \overrightarrow{\mathrm{b}}^{\prime}=\overrightarrow{\mathrm{c}} \cdot \overrightarrow{\mathrm{c}}^{\prime}=1\)

(b) \(\overrightarrow{\mathrm{a}} \cdot \overrightarrow{\mathrm{b}}^{\prime}=\overrightarrow{\mathrm{a}} \cdot \overrightarrow{\mathrm{c}}^{\prime}=0, \overrightarrow{\mathrm{b}} \cdot \overrightarrow{\mathrm{a}}^{\prime}=\overrightarrow{\mathrm{b}} \cdot \overrightarrow{\mathrm{c}}^{\prime}=0, \overrightarrow{\mathrm{c}} \cdot \overrightarrow{\mathrm{a}}^{\prime}=\overrightarrow{\mathrm{c}} \cdot \overrightarrow{\mathrm{b}}^{\prime}=0\)

(c) \([\overrightarrow{\mathrm{a}} \overrightarrow{\mathrm{b}} \overrightarrow{\mathrm{c}}]\left[\overrightarrow{\mathrm{a}}^{\prime} \overrightarrow{\mathrm{b}}^{\prime} \overrightarrow{\mathrm{c}}^{\prime}\right]=1\)

(d) \(\overrightarrow{\mathrm{a}}=\frac{\overrightarrow{\mathrm{b}} \times \overrightarrow{\mathrm{c}}^{\prime}}{\left[\overrightarrow{\mathrm{a}}^{\prime} \overrightarrow{\mathrm{b}}^{\prime} \overrightarrow{\mathrm{c}}^{\prime}\right]}, \overrightarrow{\mathrm{b}}=\frac{\overrightarrow{\mathrm{c}}^{\prime} \times \overrightarrow{\mathrm{a}}^{\prime}}{\left[\overrightarrow{\mathrm{a}}^{\prime} \overrightarrow{\mathrm{b}}^{\prime} \overrightarrow{\mathrm{c}}^{\prime}\right]}, \overrightarrow{\mathrm{c}}=\frac{\overrightarrow{\mathrm{a}}^{\prime} \times \overrightarrow{\mathrm{b}}^{\prime}}{\left[\overrightarrow{\mathrm{a}}^{\prime} \overrightarrow{\mathrm{b}}^{\prime} \overrightarrow{\mathrm{c}}^{\prime}\right]}\)Note : System of \(\hat{\mathbf{i}}, \hat{\mathbf{j}}, \hat{\mathbf{k}}\) are its own R. P. system of vector’s

(x) Bisector of an angle:The bisector of an angle between the lines \(\overrightarrow{\mathrm{r}}\) = α \(\overrightarrow{\mathrm{a}}\) and \(\overrightarrow{\mathrm{r}}\) = β\(\overrightarrow{\mathrm{b}}\) can be given by \(\overrightarrow{\mathrm{r}}\) = λ\(\left(\frac{\overrightarrow{\mathrm{a}}}{|\overrightarrow{\mathrm{a}}|} \pm \frac{\overrightarrow{\mathrm{b}}}{|\overrightarrow{\mathrm{b}}|}\right)\), λ ∈ R

(xi) Vector equation of straight line passing through any point ” \(\overrightarrow{\mathrm{a}}\)” and parallel to ” \(\overrightarrow{\mathrm{b}}\)” is given by \(\overrightarrow{\mathrm{r}}=\overrightarrow{\mathrm{a}}+\lambda \overrightarrow{\mathrm{b}}\)

(xii) Vector-equation of straight line passing through two given points ” \(\overrightarrow{\mathrm{a}}\)” and “\(\overrightarrow{\mathrm{b}}\)” is given by \(\overrightarrow{\mathbf{r}}=\overrightarrow{\mathbf{a}}+\lambda(\overrightarrow{\mathrm{b}}-\overrightarrow{\mathrm{a}})\)

(xiii) Vector equation of the plane passing through \(\overrightarrow{\mathbf{a}}\) and parallel to \(\overrightarrow{\mathbf{b}}\) and \(\overrightarrow{\mathbf{c}}\) is given by \(\overrightarrow{\mathbf{r}}=\overrightarrow{\mathbf{a}}+\lambda \overrightarrow{\mathrm{b}}+\mu \overrightarrow{\mathrm{c}}\)

(xiv) Vector equation of the plane passing through \(\overrightarrow{\mathrm{a}}, \overrightarrow{\mathrm{b}} \text { and } \overrightarrow{\mathrm{c}}\) is \(\overrightarrow{\mathrm{r}}=(1-\lambda-\mu) \overrightarrow{\mathrm{a}}+\mu \overrightarrow{\mathrm{b}}+\mu \overrightarrow{\mathrm{c}}\) where λ and µ are scalar’s.

(xv) Perpendicular distance of the line \(\overrightarrow{\mathrm{r}}=\overrightarrow{\mathrm{a}}+\lambda \overrightarrow{\mathrm{b}}\) from the point \(\mathrm{P}(\overrightarrow{\mathrm{c}})=\frac{|(\overrightarrow{\mathrm{c}}-\overrightarrow{\mathrm{a}}) \times \overrightarrow{\mathrm{b}}|}{|\overrightarrow{\mathrm{b}}|}\)

(xvi) The condition that two lines \(\overrightarrow{\mathrm{r}}=\overrightarrow{\mathrm{a}}+\lambda \overrightarrow{\mathrm{b}}\) and \(\overrightarrow{\mathrm{r}}=\overrightarrow{\mathrm{c}}+\mu \overrightarrow{\mathrm{d}}\) are COPLANER, is given by \([\vec{a}-\vec{c} \vec{b} \vec{d}]=0\).

(xvii) The shortest distance between two skew lines (non intersecting lines) can be given by \(\frac{[\vec{b}\overrightarrow{\mathrm{d}}(\overrightarrow{\mathrm{a}}-\overrightarrow{\mathrm{c}})]}{|\overrightarrow{\mathrm{b}} \times \overrightarrow{\mathrm{d}}|}\)

(xviii) Vector equation of sphere with centre \(\overrightarrow{\mathrm{a}}\) and radius p is \(|\overrightarrow{\mathrm{r}}-\overrightarrow{\mathrm{a}}|=\mathrm{p}\)

(xix) Vector equation of sphere when extremities of diameter being \(\overrightarrow{\mathrm{a}}, \overrightarrow{\mathrm{b}}\) is given by \((\overrightarrow{\mathrm{r}}-\overrightarrow{\mathrm{a}}) \cdot(\overrightarrow{\mathrm{r}}-\overrightarrow{\mathrm{b}})=0\).

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