a, b, c are non-coplanar vectors. If a+3 b+4 | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(B) Given the equation: $a+3 b+4 c=x(a-2 b+3 c)+y(a+5 b-2 c)+z(6 a+14 b+4 c)$.Expanding the right side,we get: $a+3 b+4 c=a(x+y+6 z)+b(-2 x+5 y+14 z)+

Vedclass · Accepted Answer

$-4$

Vedclass · Answer

(B) Given the equation: $a+3 b+4 c=x(a-2 b+3 c)+y(a+5 b-2 c)+z(6 a+14 b+4 c)$.Expanding the right side,we get: $a+3 b+4 c=a(x+y+6 z)+b(-2 x+5 y+14 z)+c(3 x-2 y+4 z)$.Since $a, b, c$ are non-coplanar vectors,their coefficients must be equal on both sides.Comparing the coefficients,we obtain the system of linear equations:$x+y+6 z=1$ ...$(i)$$-2 x+5 y+14 z=3$ ...(ii)$3 x-2 y+4 z=4$ ...(iii)Solving these equations:From $(i)$,$x = 1 - y - 6z$.Substituting into (ii): $-2(1-y-6z) + 5y + 14z = 3 \implies -2 + 2y + 12z + 5y + 14z = 3 \implies 7y + 26z = 5$ ...(iv).Substituting into (iii): $3(1-y-6z) - 2y + 4z = 4 \implies 3 - 3y - 18z - 2y + 4z = 4 \implies -5y - 14z = 1$ ...$(v)$.Solving (iv) and $(v)$: Multiply (iv) by $5$ and $(v)$ by $7$: $35y + 130z = 25$ and $-35y - 98z = 7$.Adding them: $32z = 32 \implies z = 1$.Substituting $z=1$ into $(v)$: $-5y - 14(1) = 1 \implies -5y = 15 \implies y = -3$.Substituting $y=-3, z=1$ into $(i)$: $x - 3 + 6(1) = 1 \implies x + 3 = 1 \implies x = -2$.Thus,$x+y+z = -2 - 3 + 1 = -4$.

$a, b, c$ are non-coplanar vectors. If $a+3 b+4 c=x(a-2 b+3 c)+y(a+5 b-2 c)+z(6 a+14 b+4 c)$,then $x+y+z=$

Similar Questions

$(r \cdot i)^2 + (r \cdot j)^2 + (r \cdot k)^2 = $

The value of $k$ for which the vectors $\vec{a} = \hat{i} - \hat{j}$ and $\vec{b} = -2\hat{i} + k\hat{j}$ are collinear is

If $\vec{a}=-2 \hat{i}+9 \hat{j}-6 \hat{k}$ and $\vec{b}=t \hat{i}-2 \hat{j}+6 \hat{k}$ are vectors such that $|\vec{a}+\vec{b}|=25$,then the sum of the values of $t$ is

The projection of $\hat{i}-\hat{j}$ on $\hat{i}+\hat{j}$ is . . . . . . .

Five points given by $A, B, C, D, E$ are in a plane. Three forces $\overrightarrow{AC}, \overrightarrow{AD},$ and $\overrightarrow{AE}$ act at $A$,and three forces $\overrightarrow{CB}, \overrightarrow{DB},$ and $\overrightarrow{EB}$ act at $B$. Then their resultant is:

Vedclass Test Series

Exam Paper Generator

Online Exam Module