If the projection of the vector hat i +2 hat j + hat k on the sum of the two vectors 2 hat i +4 hat j -5 hat k and -λ hat i +2 hat j +3 hat k is 1 , then λ is equal to

Question

If the projection of the vector hat i +2 hat j + hat k on the sum of the two vectors 2 hat i +4 hat j -5 hat k and -λ hat i +2 hat j +3 hat k is 1 , then λ is equal to  (A)  (B)  (C)  (D)

Tardigrade Admin · Accepted Answer

vec a = hat i +2 hat j + hat k vec b =(2-λ) hat i +6 hat j -2 hat k ( vec a ⋅ vec b /| vec b |)=1, vec a ⋅ vec b =12-λ ( vec a ⋅ vec b )=| vec b |2 λ2-24 λ+144 =λ2-4 λ+4+40 20 λ=100 ⇒ λ=5

Q. If the projection of the vector $\hat{i} + 2 \hat{j} + \hat{k}$ on the sum of the two vectors $2 \hat{i} + 4 \hat{j} - 5 \hat{k}$ and $- λ \hat{i} + 2 \hat{j} + 3 \hat{k}$ is $1$ , then $λ$ is equal to ___

$a = \hat{i} + 2 \hat{j} + \hat{k}$
$b = (2 - λ) \hat{i} + 6 \hat{j} - 2 \hat{k}$
$\frac{a \cdot b}{∣ b ∣} = 1, a \cdot b = 12 - λ$
$(a \cdot b) = ∣ b ∣^{2}$
$λ^{2} - 24 λ + 144$
$= λ^{2} - 4 λ + 4 + 40$
$20 λ = 100$
$\Rightarrow λ = 5$

Q. If the projection of the vector i^+2j^​+k^ on the sum of the two vectors 2i^+4j^​−5k^ and −λi^+2j^​+3k^ is 1 , then λ is equal to ___

a=i^+2j^​+k^ b=(2−λ)i^+6j^​−2k^ ∣b∣a⋅b​=1,a⋅b=12−λ (a⋅b)=∣b∣2 λ2−24λ+144 =λ2−4λ+4+40 20λ=100 ⇒λ=5

Q. If the projection of the vector $\hat{i} + 2 \hat{j} + \hat{k}$ on the sum of the two vectors $2 \hat{i} + 4 \hat{j} - 5 \hat{k}$ and $- λ \hat{i} + 2 \hat{j} + 3 \hat{k}$ is $1$ , then $λ$ is equal to ___

$a = \hat{i} + 2 \hat{j} + \hat{k}$
$b = (2 - λ) \hat{i} + 6 \hat{j} - 2 \hat{k}$
$\frac{a \cdot b}{∣ b ∣} = 1, a \cdot b = 12 - λ$
$(a \cdot b) = ∣ b ∣^{2}$
$λ^{2} - 24 λ + 144$
$= λ^{2} - 4 λ + 4 + 40$
$20 λ = 100$
$\Rightarrow λ = 5$