If the moduli of vectors a, b, c are 3,4,5 are respectively and a and b+c, b and c+a, c and a+b are mutually perpendicular then the modulus of a+b+c is

Question

If the moduli of vectors a, b, c are 3,4,5 are respectively and a and b+c, b and c+a, c and a+b are mutually perpendicular then the modulus of a+b+c is (A) √12 (B) 12 (C) 5 √2 (D) 50

Tardigrade Admin · Accepted Answer

According to the given condition, we have

a \cdot (b + c) = 0 (1)

b \cdot (c + a) = 0 (2)

c \cdot (a + b) = 0 (3)

Now adding (1), (2) and (3), we get

2 (a \cdot b + b \cdot c + c \cdot a) = 0

(∵ a \cdot b = b \cdot a

etc.

)

Hence,

∣ a + b + c ∣^{2}

= a^{2} + b^{2} + c^{2} + 2

(a \cdot b + b \cdot c + c \cdot a)

= 3^{2} + 4^{2} + 5^{2}

= 9 + 16 + 25 = 50

\Rightarrow ∣ a + b + c ∣

= 50 = 52

Q. If the moduli of vectors $a, b, c$ are $3, 4, 5$ are respectively and $a$ and $b + c, b$ and $c + a, c$ and $a + b$ are mutually perpendicular then the modulus of $a + b + c$ is

$12$

$12$

$52$

$50$

Q. If the moduli of vectors a,b,c are 3,4,5 are respectively and a and b+c,b and c+a,c and a+b are mutually perpendicular then the modulus of a+b+c is

12​

12

52​

50

According to the given condition, we have a⋅(b+c)=0(1) b⋅(c+a)=0(2) c⋅(a+b)=0(3) Now adding (1), (2) and (3), we get 2(a⋅b+b⋅c+c⋅a)=0 (∵a⋅b=b⋅a etc. ) Hence, ∣a+b+c∣2 =a2+b2+c2+2 (a⋅b+b⋅c+c⋅a) =32+42+52 =9+16+25=50 ⇒∣a+b+c∣ =50​=52​

Q. If the moduli of vectors $a, b, c$ are $3, 4, 5$ are respectively and $a$ and $b + c, b$ and $c + a, c$ and $a + b$ are mutually perpendicular then the modulus of $a + b + c$ is

$12$

$12$

$52$

$50$