If 𝐫=⟨x, y, z⟩, 𝐚=⟨a1, a2, a3⟩, and 𝐛=⟨b1, b2, b3⟩show that the vector equation (𝐫-𝐚) ·(𝐫-𝐛)=0 r

step 1 R minus A into R minus B equals 0. Therefore it is R minus A into R minus B equal R into R minus

If 𝐫=⟨x, y, z⟩, 𝐚=⟨a1, a2, a3⟩, and 𝐛=⟨b1, b2, b3⟩show that...

If $\mathbf{r}=\langle x, y, z\rangle, \mathbf{a}=\left\langle a_{1}, a_{2}, a_{3}\right\rangle,$ and $\mathbf{b}=\left\langle b_{1}, b_{2}, b_{3}\right\rangle$ show that the vector equation $(\mathbf{r}-\mathbf{a}) \cdot(\mathbf{r}-\mathbf{b})=0$ represents a sphere, and find its center and radius.

If $\mathbf{r}=\langle x, y, z\rangle, \mathbf{a}=\left\langle a_{1}, a_{2}, a_{3}\right\rangle,$ and $\mathbf{b}=\left\langle b_{1}, b_{2}, b_{3}\right\rangle$
show that the vector equation $(\mathbf{r}-\mathbf{a}) \cdot(\mathbf{r}-\mathbf{b})=0$ represents a sphere, and find its center and radius.

Key Concepts

Sphere Equation in Coordinate Geometry

A sphere in three-dimensional space is defined as the set of all points that are at a fixed distance (radius) from a given point (center). The standard form of its equation is (x - h)² + (y - k)² + (z - l)² = r², where (h, k, l) is the center and r is the radius. Recognizing and deriving this form from a given vector equation is an important skill in analytic geometry.

Quadratic Forms and Completing the Square

Quadratic equations in vector form often lead to expressions that include squared terms of the coordinates. Completing the square is a method used to rewrite such expressions in a form that clearly shows the center and the radius of a sphere. This technique is critical in transforming the expanded vector equation into the standard form of a sphere's equation.

Dot Product

The dot product is an algebraic operation that takes two equal-length sequences of numbers (usually coordinate vectors) and returns a single number. It is used to measure the extent to which two vectors are pointing in the same direction. Its properties, such as distributivity and bilinearity, make it a powerful tool in forming and manipulating equations in vector algebra.

Vector Equation Representation

Vector equations involve expressions that use vectors to represent geometric objects. Here, the equation (r - a) · (r - b) = 0 is a condition imposed on the position vector r, and by manipulating it, one can derive a familiar geometric form. Such representations help in visualizing and understanding properties like distance and orthogonality in three-dimensional space.

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