find the lamda, if a + lamda b is parallel to the vector i?

Question

Given that a = 2i + j and b= i + 3 j

Please show the calculations .

Answer 1

Vector A and Vector B are parallel then their cross product is zero
A cross B = |A| |B| sin (0) n-hat (unit vector)
= 0 (as sin 0 = 0)

[(2i + j) + lamda* (i + 3 j)] cross [ i ] = 0

take cross product
2 (i X i) +(j X i) + lamda*[(i X i) + 3 (j X i)] = 0
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(i X i) = 0 (sin o) (j X i) = - (i X j) = - k
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(- k) + lamda*[ - 3 (k)] = 0

k + lamda*[ 3 k] = 0
compare the coefficients of k on both sides

1 + 3 lamda = 0 >>> lamda = - 1/3
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dot product

A dot B = |A | |B| cos (0) = |A | |B| (as cos 0 = 1)

A = (2+lamda) i + (1+3 lamda) j
|A | = sqrt [(2+lamda)^2 + (1+3 lamda)^2]
|B | =1

(2+lamda) i + (1+3 lamda) j] dot [ i ] =
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i dot i = 1, i dot j =0
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(2+lamda) +o = sqrt [(2+lamda)^2 + (1+3 lamda)^2] [1]
squaring
(2+lamda)^2 = [(2+lamda)^2 + (1+3 lamda)^2]

0 = (1+3 lamda)^2 >>> lamda = - 1/3

Answer 2

a and b are beside the point as long as they're actual numbers, wherein case (2i+3j) || a(2i+3j) || b(3i + lambda j) || (3i + lambda j). So we want lambda such that (2 i + 3 j) || (3 i + lambda j). The coefficients could be in share, so 2 / 3 = 3 /lambda --> lambda = 9/2.

Answer 3

Given vectors a and b.

a = 2i + j
b= i + 3 j

Find λ such that a + λb = ki, where λ and k are scalars.

a + λb = (2i + j) + λ(i + 3j) = (2 + λ)i + (1 + 3λ)j = ki + 0j

1 + 3λ = 0
3λ = -1
λ = -1/3