Question

If a ⃗=(0,1 ,-1) and c ⃗= (1, 1, 1) are given vectors, then find a vector b ⃗ satisfying a ⃗×b ⃗+c ⃗ = 0 and a ⃗ .b ⃗ = 3.

Answer 1

Answer on Question #44080, Math, Vector Calculus

If $a \to (0,1,-1)$ and $c \to (1,1,1)$ are given vectors, then find a vector $b \to$ satisfying $a \to \mathbf{b} \to c \to 0$ and $a \to .b \to 3$.

Solution.

Assume that vector $\vec{b}$ has coordinates $(x,y,z)$. Then $\vec{a} \times \vec{b} + \vec{c} = \begin{vmatrix} i & j & k \\ 0 & 1 & -1 \\ x & y & z \end{vmatrix} + (1,1,1) = i(z + y) - jx - kx + (1,1,1) = (1 + y + z, 1 - x, 1 - x) = (0,0,0)$.

Also, $\vec{a} \cdot \vec{b} = 0 \cdot x + 1 \cdot y - 1 \cdot z = y - z = 3$.

Now we solve the system of equations:

$$\left\{ \begin{array}{l} 1 + y + z = 0, \\ 1 - x = 0, \\ y - z = 3; \end{array} \right. \left\{ \begin{array}{l} 1 + (z + 3) + z = 0, \\ x = 1, \\ y = z + 3; \end{array} \right. \left\{ \begin{array}{l} z = -2, \\ x = 1, \\ y = 1. \end{array} \right.$$

Hence, $\vec{b} = (1,1,-2)$.

Answer: $(1,1,-2)$

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