Find the vector \(\mathbf{v}\) such that \(\mathbf{v} \times \mathbf{b} = \langle 1, 0, -1 \rangle\) and \(\mathbf{b} = \langle 0, 1, 0 \rangle\).
["Finding Vector (\mathbf{v}) Such That (\mathbf{v} \ imes \mathbf{b} = \langle 1, 0, -1 \rangle) With (\mathbf{b} = \langle 0, 1, 0 \rangle)", "When solving vector cross product equations like (\mathbf{v} \ imes \mathbf{b} = \mathbf{c}), where (\mathbf{b}) and (\mathbf{c}) are known vectors, the solution for (\mathbf{v}) often exhibits multiple possibilities due to the nature of cross products. In this article, we will find all vectors (\mathbf{v} = \langle x, y, z \rangle) such that\n[\n\mathbf{v} \ imes \langle 0, 1, 0 \rangle = \langle 1, 0, -1 \rangle.\n]", "---", "### Understanding the Cross Product", "Given (\mathbf{v} = \langle x, y, z \rangle) and (\mathbf{b} = \langle 0, 1, 0 \rangle), the cross product is computed as:", "[\n\mathbf{v} \ imes \mathbf{b} = \n\begin{vmatrix}\n\mathbf{i} & \mathbf{j} & \mathbf{k} \\nx & y & z \\n0 & 1 & 0\n\end{vmatrix}\n= \mathbf{i}(y \cdot 0 - z \cdot 1) - \mathbf{j}(x \cdot 0 - z \cdot 0) + \mathbf{k}(x \cdot 1 - y \cdot 0)\n= \langle -z, 0, x \rangle.\n]", "We require:\n[\n\langle -z, 0, x \rangle = \langle 1, 0, -1 \rangle.\n]", "Equating components:\n- (-z = 1 \Rightarrow z = -1)\n- (x = -1)", "The (y)-component is already zero in both vectors, so (y) remains free — meaning any real number for (y) satisfies the equation.", "---", "### General Solution", "From the above, we conclude:\n[\n\mathbf{v} = \langle x, y, z \rangle = \langle -1, y, -1 \rangle, \quad y \in \mathbb{R}.\n]", "Thus, the set of all solutions is the line in 3D space where the (x)-component is (-1), the (z)-component is (-1), and the (y)-component is unrestricted.", "---", "### Why Multiple Solutions?", "The cross product (\mathbf{v} \ imes \mathbf{b}) depends only on the components of (\mathbf{v}) orthogonal to (\mathbf{b}). Since (\mathbf{b} = \langle 0, 1, 0 \rangle), the result only involves (x) and (z), leaving (y) unconstrained. This is why no unique solution exists — instead, there’s a one-dimensional family of solutions.", "---", "### Final Answer", "All vectors (\mathbf{v}) satisfying (\mathbf{v} \ imes \langle 0, 1, 0 \rangle = \langle 1, 0, -1 \rangle) are given by:", "[\n\boxed{\mathbf{v} = \langle -1, y, -1 \rangle \quad \ ext{for any real number } y}\n]", "This line-based solution is elegant, exact, and optimal for representing the full set of vectors that produce the desired cross product.", "---", "### SEO Keywords:\nvector cross product solution, find vector v such that v × b = ⟨1,0,−1⟩, solve v × b = ⟨1,0,−1⟩, v cross b equals ⟨1,0,−1⟩, vector equation solution, parametric vector, unsolvable cross product components, mathematical vector solution, 3D vector algebra.", "---", "For more insights on cross products and vector equations, explore our full guide on linear algebra fundamentals."]
