Chapter 10
is a vector. 2. Let be two nonzero vectors. Then if and only if are parallel (or collinear) to each other, i.e., = MATHEMATICS364 In particular , and , since in the first situation, θ = 0 and in the second one, θ = π , making the value of si n θ to be 0. 3. If πθ = then . 4. In view of the Observations 2 and 3, for mutually perpendicular unit vectors ˆˆ ˆ, andi j k (Fig 10.24), we have ˆ ˆi i× = ˆ ˆi j× = ˆ ˆ ˆ ˆ ˆ ˆ ˆ, , k j k i k i j× = × = 5. In terms of vector product, the angle between two vector s may be given as sin θ = 6. It is always true that the vector product is not commutative, as = . Indeed, , where form a right handed system, i.e., θ is traversed from , Fig 10.25 (i). While, , where form a right handed system i.e. θ is traversed from , Fig 10.25(ii). Fig 10.25 (i), (ii) Thus, if we assume to lie in the plane of the paper, then 1ˆ ˆ and n n both will be perpendicular to the plane of the paper. But, ˆn being directed above the paper while 1ˆn directed below the paper. i.e. 1ˆ ˆn n= − . Hence = = 7. In view of the Observations 4 and 6, we have ˆ ˆ ˆˆ ˆ ˆ ˆ ˆ ˆ , and .j i k k j i i k j× = − × = − × = − 8. If represent the adjacent sides of a triangle then its area is given as . By definition of the area of a triangle, we have from Fig 10.26, Area of triangle ABC = 1 AB CD.2 ⋅ But (as given), and CD = sin θ . Thus, Area of triangle ABC = 9. If represent the adjacent sides of a parallelogram, then its area is given by . From Fig 10.27, we have Area of parallelogram ABCD = AB. DE. But (as given), and . Thus, Area of parallelogram ABCD = We now state two important properties of vector product. Property 3 (Distributivity of vector product over addition): If are any three vectors and λ be a scalar , then (i) (ii) MATHEMATICS366
| Let be two vectors given in component form as | 1 | 2 | 3 |
|---|
ˆˆ ˆa i a j a k+ + and 1 2 3 ˆˆ ˆb i b j b k+ + , respectively. Then their cross product may be given by = 1 2 3 1 2 3 ˆˆ ˆi j k
| a | a | a |
|---|---|---|
| b | b | b |
Explanation We have = 1 2 3 1 2 3 ˆ ˆˆ ˆ ˆ ˆ( ) ( )a i a j a k b i b j b k+ + × + + = 1 1 1 2 1 3 2 1 ˆˆ ˆ ˆ ˆ ˆ ˆ ˆ( ) ( ) ( ) ( )a b i i a b i j a b i k a b j i× + × + × + × + 2 2 2 3 ˆˆ ˆ ˆ( ) ( )a b j j a b j k× + × + 3 1 3 2 3 3 ˆ ˆ ˆ ˆˆ ˆ( ) ( ) ( )a b k i a b k j a b k k× + × + × (by Property 1) = 1 2 1 3 2 1 ˆˆ ˆ ˆ ˆ ˆ( ) ( ) ( )a b i j a b k i a b i j× − × − × + 2 3 3 1 3 2 ˆ ˆ ˆˆ ˆ ˆ( ) ( ) ( )a b j k a b k i a b j k× + × − × ˆ ˆ ˆ ˆ ˆ ˆˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ(as 0 and , and )i i j j k k i k k i j i i j k j j k× = × = × = × = − × × = − × × = − × = 1 2 1 3 2 1 2 3 3 1 3 2 ˆ ˆˆ ˆ ˆ ˆa b k a b j a b k a b i a b j a b i− − + + − ˆ ˆ ˆˆ ˆ ˆ ˆ ˆ ˆ(as , and )i j k j k i k i j× = × = × = = 2 3 3 2 1 3 3 1 1 2 2 1 ˆˆ ˆ( ) ( ) ( )a b a b i a b a b j a b a b k− − − + − = 1 2 3 1 2 3 ˆˆ ˆi j k
| a | a | a |
|---|---|---|
| b | b | b |
Example 22 Find Solution We have = ˆˆ ˆ 2 1 3 3 5 2
| i | j | k |
|---|---|---|
| − = ˆˆ ˆ( 2 15) ( 4 9) (10 - 3)i j k− − − − − + ˆˆ ˆ17 13 7i j k= − + + Hence = 2 2 2( 17) (13) (7) 507− + + = Example 23 Find a unit vector perpendicular to each of the vectors and where . Solution We have A vector which is perpendicular to both and+ − /arrowrightnosp /arrowrightnosp/arrowrightnosp /arrowrightnospa b a b is given by = Now = 4 16 4 24 2 6+ + = = Therefore, the required unit vector is | /arrowrightnosp /arrowrightnospc c = 1 2 1 ˆˆ ˆ 6 6 6 i j k− + − /handptrtsld1Note There are two perpendicular directions to any plane. Thus, another unit vector perpendicular to will be 1 2 1 ˆˆ ˆ . 6 6 6 i j k− + But that will be a consequence of . Example 24 Find the area of a triangle having the points A(1, 1, 1), B(1, 2, 3) and C(2, 3, 1) as its vertices. Solution We have . The area of the given triangle is . Now, = ˆˆ ˆ ˆˆ ˆ0 1 2 4 2 1 2 0 | |
| i | j | k |
| --- | --- | --- |
i j k= − + − Therefore = 16 4 1 21+ + = Thus, the required area is 1 21 MATHEMATICS368 Example 25 Find the area of a parallelogram whose adjacent sides are given by the vectors Solution The area of a parallelogram with as its adjacent sides is given by . Now = ˆˆ ˆ ˆˆ ˆ3 1 4 5 4 1 1 1
| i | j | k |
|---|
i j k= + − − Therefore = 25 1 16 42+ + = and hence, the required area is 42 .
If are such that is perpendicular to , then find the value of λ .
Vertices (± 5, 0), foci (± 4, 0)
Show that is perpendicular to , for any two nonzero vectors .
Vertices (0, ± 13), foci (0, ± 5)
If , then what can be concluded about the vector ?
Vertices (± 6, 0), foci (± 4, 0)
Ends of major axis (± 3, 0), ends of minor axis (0, ± 2)
If are unit vectors such that , find the value of .
If either vector . But the converse need not be true. Justify your answer with an example.
Ends of major axis (0, ± 5 ), ends of minor axis (± 1, 0)
Length of major axis 26, foci (± 5, 0)
If the vertices A, B, C of a triangle ABC are (1, 2, 3), (-1, 0, 0), (0, 1, 2), respectively, then find ∠ ABC. [∠ ABC is the angle between the vectors and ].
Show that the points A(1, 2, 7), B(2, 6, 3) and C(3, 10, -1) are collinear.
Length of minor axis 16, foci (0, ± 6).
Show that the vectors ˆ ˆ ˆˆ ˆ ˆ ˆ ˆ ˆ2 , 3 5 and 3 4 4i j k i j k i j k− + − − − − form the vertices of a right angled triangle.
Foci (± 3, 0), a = 4
b = 3, c = 4, centre at the origin; foci on the x axis.
If is a nonzero vector of magnitude ‘a’ and λ a nonzero scalar, then λ is unit vector if (A) λ = 1 (B) λ = - 1 (C) a = | λ | (D) a = 1/| λ | 10.6.3 Vector (or cross) product of two vectors In Section 10.2, we have discussed on the three dimensional right handed rectangular coordinate system. In this system, when the positive x-axis is rotated counterclockwise into the positive y-axis, a right handed (standard) screw would advance in the direction of the positive z-axis (Fig 10.22(i)). In a right handed coordinate system, the thumb of the right hand points in the direction of the positive z-axis when the fingers are curled in the direction away from the positive x-axis toward the positive y-axis (Fig 10.22(ii)). Fig 10.22 (i), (ii) Definition 3 The vector product of two nonzero vectors , is denoted by and defined as = , where, θ is the angle between , 0 ≤ θ ≤ π and ˆn is a unit vector perpendicular to both , such that form a right handed system (Fig 10.23). i.e., the right handed system rotated from moves in the direction of ˆn . If either , then θ is not defined and in this case, we define . Observations
Centre at (0,0), major axis on the y-axis and passes through the points (3, 2) and (1,6).
Find the angle between the vectors ˆ ˆˆ ˆ ˆ ˆ2 3 and 3 2i j k i j k− + − + and ˆ ˆˆ ˆ ˆ ˆ2 3 and 3 2i j k i j k− + − +
Major axis on the x-axis and passes through the points (4,3) and (6,2).
Find the projection of the vector ˆ ˆi j− on the vector ˆ ˆi j+ .
Find the projection of the vector ˆˆ ˆ 3 7i j k+ + on the vector ˆˆ ˆ7 8i j k− + .
Show that each of the given three vectors is a unit vector: 1 1 1ˆ ˆ ˆˆ ˆ ˆ ˆ ˆ ˆ(2 3 6 ), (3 6 2 ), (6 2 3 )7 7 7i j k i j k i j k+ + − + + − Also, show that they are mutually perpendicular to each other. MATHEMATICS362
Find , if .
Evaluate the product .
36x2 + 4y2 = 144 8. 16x2 + y2 = 16 9. 4x2 + 9y2 = 36 In each of the following Exercises 10 to 20, find the equation for the ellipse that satisfies the given conditions:
Find the magnitude of two vectors , having the same magnitude and such that the angle between them is 60o and their scalar product is 1 .
Find , if for a unit vector , .