Show that points P(6,−2,−3), Q(12,−5,−6), and R(8,−3,−4) are collinear.

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ScanSolve AI Tutor · Accepted Answer

To show that points P, Q, and R are collinear, we need to demonstrate that two vectors formed by these points (e.g., $\vec{PQ}$ and $\vec{QR}$ ) are parallel and share a common point. Parallel vectors are scalar multiples of each other.

Express the position vectors in column form.
Given position vectors are:
$\vec{OP} = 6\mathbf{i} - 2\mathbf{j} - 3\mathbf{k} = \begin{pmatrix} 6 \\ -2 \\ -3 \end{pmatrix}$ $\vec{OQ} = 12\mathbf{i} - 5\mathbf{j} - 6\mathbf{k} = \begin{pmatrix} 12 \\ -5 \\ -6 \end{pmatrix}$
Due to the blurriness in the image for $\vec{OR}$ , and the requirement to "show that P, Q and R are collinear", we assume the last component of $\vec{OR}$ is $-4\mathbf{k}$ for collinearity to hold.
$\vec{OR} = 8\mathbf{i} - 3\mathbf{j} - 4\mathbf{k} = \begin{pmatrix} 8 \\ -3 \\ -4 \end{pmatrix}$
Calculate the vectors $\vec{PQ}$ and $\vec{QR}$ .
$\vec{PQ} = \vec{OQ} - \vec{OP} = \begin{pmatrix} 12 \\ -5 \\ -6 \end{pmatrix} - \begin{pmatrix} 6 \\ -2 \\ -3 \end{pmatrix} = \begin{pmatrix} 12-6 \\ -5-(-2) \\ -6-(-3) \end{pmatrix} = \begin{pmatrix} 6 \\ -3 \\ -3 \end{pmatrix}$ $\vec{QR} = \vec{OR} - \vec{OQ} = \begin{pmatrix} 8 \\ -3 \\ -4 \end{pmatrix} - \begin{pmatrix} 12 \\ -5 \\ -6 \end{pmatrix} = \begin{pmatrix} 8-12 \\ -3-(-5) \\ -4-(-6) \end{pmatrix} = \begin{pmatrix} -4 \\ 2 \\ 2 \end{pmatrix}$
Check if $\vec{PQ}$ is a scalar multiple of $\vec{QR}$ .
We look for a scalar $\lambda$ such that $\vec{PQ} = \lambda \vec{QR}$ .
Comparing the components:
For the $\mathbf{i}$ component: $6 = \lambda(-4) \implies \lambda = \frac{6}{-4} = -\frac{3}{2}$
For the $\mathbf{j}$ component: $-3 = \lambda(2) \implies \lambda = -\frac{3}{2}$
For the $\mathbf{k}$ component: $-3 = \lambda(2) \implies \lambda = -\frac{3}{2}$
Since $\lambda = -\frac{3}{2}$ is consistent for all components, we have $\vec{PQ} = -\frac{3}{2} \vec{QR}$ .
Conclude collinearity.
Since $\vec{PQ}$ is a scalar multiple of $\vec{QR}$ , the vectors are parallel. As they share a common point Q, the points P, Q, and R lie on the same straight line.
Therefore, P, Q, and R are collinear points.

Show that points P(6,-2,-3), Q(12,-5,-6), and R(8,-3,-4) are collinear.

Step-by-step solution

Express the position vectors in column form.

Calculate the vectors $\vec{PQ}$ and $\vec{QR}$ .

Check if $\vec{PQ}$ is a scalar multiple of $\vec{QR}$ .

Conclude collinearity.

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