Vector Algebra Question?

Question

Question:If (a)i +(2b)j+(a)k and (2b-c)i +(3a+c)j+(b)k are parallel, then find the unit vector in their direction.

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My Solution:-Since the two vectors are parallel a/(2b-c)=2b/(3a+c)=a/b=2(a+b)/(3a+3b)=2/3

Or, a= 2/3b ….. now how to solve further.Please Help.

a/(2b-c)=2b/(3a+c)=a/b=2(a+b)/(3a+3b)=2/3
Or, a= 2/3b

Answer 1

You've started out right.

a/(2b - c) = 2b/(3a + c) = a/b

Take the first and third expressions.

a/(2b - c) = a/b
2b - c = b
b = c

Plug back in and we have:

a/(2c - c) = 2c/(3a + c) = a/c
a/c = 2c/(3a + c) = a/c

a/c = 2c/(3a + c)
a(3a + c) = 2c²
3a² + ac - 2c² = 0
(3a - 2c)(a + c) = 0
a = 2c/3, -c

Now we have:

-c/(2c - c) = 2c/(-3c + c) = -c/c
-c/c = 2c/(-2c) = -c/c

which checks.

Now plug into the first vector.

ai + 2bj + ak = -ci + 2cj - ck

The magnitude of the vector is:

√[(-c)² + (2c)² + (-c)²] = √(6c²) = c√6

The unit vector in their direction is:

<-ci + 2cj - ck> / (c√6) = <-1, 2, -1> / √6

Answer 2

The two vectors: (a)i + (2b)j + (a)k and (2b - c)i + (3a + c)j + (b)k will be parallel if:

a/(2b - c) = 2b/(3a + c) = a/b................(1)

Taking Ist & last equations: a/(2b - c) = a/b

i.e. (2b - c) = b or b = c .......................(2)

Put this in Ist & 2nd equations:

a/b = 2b/(3a + b) or 3a^2 + ab - 2b^2 = 0

or (a + b)(3a - 2b) = 0. Hence either a = - b or a = 2b/3.

Now first take a = - b, then b = b and c = b from (2) i.e.

a : b : c :: - 1 : 1 : 1 ...........................(3).

Next taking a = 2b/3, b = b, c = b from (2)

i.e. a : b : c :: 2/3 : 1 : 1 :: 2 : 3 : 3 .............(4)

From (3) and (4) either:

(a)i + (2b)j + (a)k = - i + 2j - k and i - 2j + k which are obviously parallel. And Unit Vector is obtained by dividing the vector by its magnitude or modulus = sq. root (1 +4 + 1) or

(i - 2j + k)/sq. root (6)........................(5)

OR (a)i + (2b)j + (a)k = i + 3j + k.

Again Unit Vector is obtained by dividing it by its magnitude or modulus = sq. root (1 + 9 + 1) or

(i + 3j + k)/sq. root (11).......................(6)

(5) and (6) are the desired unit vectors.

Answer 3

one million) subtract PD from the two facets: AP + PB = -PD + PD AB = -PD + laptop AB = DP + laptop AB = DC So opposites facets are congruent and parallel. 2) "Taken so as" in basic terms skill "taken so as that the vectors guidelines are cyclic", which in basic terms skill they upload to 0. in case you like a vector algebra rationalization, evaluate the vertices as vector displacements from some beginning. If vertex A is contrary facet a, and the vertex order is going A, B, C, then the vector sum is in basic terms a + b + c = (C - B) + (A - C) + (B - A) = 0. in case you like, replace A, B, C with OA, OB, OC, the place O is an arbitrary beginning element.