1. Show that a × (b × c), b × (c × a) and c × (a × b) are coplanar.
Proof: a × (b × c) = (a . c) b - (a . b)c → (1)
b × (c × a) = (b . a) c - (b . c)a → (2)
c × (a × b) = (c . b) a - (c . a)b → (3)
(1) + (2) + (3)
a × (b × c) + b × (c × a) + c × (a × b) = 0
a × (b × c), b × (c × a), c × (a × b) are coplanar
2. Find the value of i × (j × k) + j × (k × i) + k × (i × j)
Sol. Given i × (j × k) + j × (k × i) + k × (i × j)
= i × i + j × j + k × k
= 0 + 0 + 0
= 0
(or)
= (i.k)j - (i.j)k + (j.i)k - (j.k)i + (k.j)i - (k.i)j
= (0)j - (0)k + (0)k - (0)i + (0)i - (0)j
= 0
3. If = i - 2j + 3k, = 2i + j - k and = i + j + 2k then find the value of (a × b) × c
4. Show that [a × b b × c c × a] = [abc]2
Sol. L.H.S [a × b b × c c × a]
= (a × b) . ((b × c) × (c × a))
Let b × c = p
= (a × b) . (p × (c × a))
= (a × b) ((p . a) c - (p . c) a)
= (a × b) . ((b × c . a) c - (b × c . c) a)
= (a × b) . ([abc] c - [cbc] a)
= (a × b) . ([abc] c - 0)
= (a × b) . c [abc]
= [abc] [abc]
= [abc]2
= R.H.S
5. If = 2i + 3j + 4k, = i + j - k, = i - j + k then verify that a × (b × c) is perpendicular to .
Sol. Given = 2i + 3j + 4k
= i + j - k
= i - j + k
6. If i, j, k are orthogonal unit vector kind and for any vector , show that
7. Show that d . [a × {b × (c × d)}] = (b . d) [acd]
Sol: L.H.S d . [ a × {b × (c × d)}]
= d . [ a × {(b . d) c - (b . c) d}] ( By using Conceptual Theorem (1) )
= d . [(a × c)(b . d) - (a × d)(b . c)]
= d . (a × c)(b . d) - d (a × d) (b . c)
= [ d a c ] (b . d) - [d a d] (b . c)
= [ a c d ] (b . d) - 0
= (b . d) [a c d]
8. Show that (a × b) . (c × d) = (a . c) (b . d) - (a . d) (b . c) =
= (a . c) (b . d) - (a . d) (b . c) --→ (2)
From (1) and (2) The given statement is true.
9. Show that (a × b) × (c × d) = [a b d] c - [a b c] d = [a c d] b - [b c d] a
10. Show that (b × c) . (a × d) + (c × a) . (b × d) + (a × b) . (c × d) = 0
11. If = i + 2j - k, = 3i - 4k, = -i + j and = 2i - j + 3k then find
(i) (a × b) . (c × d) (ii) (a × b) × (c × d)
Sol: Given = i + 2j - k
= 3i - 4k
= -i + j
= 2i - j + 3k
(ii) (a × b) × (c × d) = - 35 (-i + j) - 9 (2i - j + 3k)
= 35i - 35j - 18i + 9j - 27k
= 17i - 26j - 27k
Writer: Sayyad Anwar