Question

The points \(A=(1,1,3), B=(3,2,7), C=(2,0,8)\) and \(D=(0,-1,4)\) form a quadrilateral \(A B C D\) in space. Is this a parallelogram?

Short answer

Yes, the quadrilateral is a parallelogram.

Step-by-step solution

Find Vectors of Opposite Sides

To determine whether quadrilateral \(ABCD\) is a parallelogram, we first need to find the vectors for opposite sides. Calculate the vector \(\overrightarrow{AB}\) by subtracting the coordinates of point \(A\) from \(B\):\[\overrightarrow{AB} = (3-1, 2-1, 7-3) = (2, 1, 4).\]Similarly, compute the vector \(\overrightarrow{CD}\):\[\overrightarrow{CD} = (0-2, -1-0, 4-8) = (-2, -1, -4).\]

Compare Vectors for Parallelism

For the quadrilateral to be a parallelogram, opposite sides should be parallel, i.e., the vectors should be scalar multiples of each other.Compare \(\overrightarrow{AB} = (2, 1, 4)\) with \(\overrightarrow{CD} = (-2, -1, -4)\).Notice that:\[\overrightarrow{CD} = -1 * \overrightarrow{AB},\]hence, \(\overrightarrow{AB}\) and \(\overrightarrow{CD}\) are indeed parallel.

Find the Other Pair of Opposite Side Vectors

Next, calculate the other pair of opposite side vectors: \(\overrightarrow{BC}\) and \(\overrightarrow{DA}\).\(\overrightarrow{BC}\) is computed as:\[\overrightarrow{BC} = (2-3, 0-2, 8-7) = (-1, -2, 1).\]\(\overrightarrow{DA}\) is computed as:\[\overrightarrow{DA} = (1-0, 1+1, 3-4) = (1, 2, -1).\]

Compare the Second Pair of Vectors

Now, compare \(\overrightarrow{BC} = (-1, -2, 1)\) with \(\overrightarrow{DA} = (1, 2, -1)\).Notice that:\[\overrightarrow{BC} = -1 * \overrightarrow{DA},\]indicating that \(\overrightarrow{BC}\) and \(\overrightarrow{DA}\) are also parallel, confirming the second pair of opposite sides are parallel.

Conclusion: Check Parallel Sides

Since both pairs of opposite sides \(\overrightarrow{AB}, \overrightarrow{CD}\) and \(\overrightarrow{BC}, \overrightarrow{DA}\) are parallel, quadrilateral \(ABCD\) is a parallelogram by definition.

Key concepts

Parallelogram

A parallelogram is a four-sided figure with opposite sides that are both parallel and equal in length. In the realm of vectors, you can analyze a quadrilateral to see if it forms a parallelogram by evaluating the properties of its sides. To verify if quadrilateral \(ABCD\) is a parallelogram, we must check if its opposite sides are parallel. This involves checking if the vectors representing these sides are scalar multiples of each other, as this indicates parallelism.

By examining two pairs of opposite vectors, if both pairs are found to be parallel, the quadrilateral is indeed a parallelogram. Not only does this help in understanding the spatial relationships in geometry, but it also highlights how vectors can simplify complex concepts.

Vector Subtraction

Vector subtraction is a crucial concept when working with coordinates in space. It involves finding the direction and length of a new vector by subtracting the components of one vector from another. In the example given, vector subtraction allows you to find the vectors of the sides of the quadrilateral \(ABCD\).

To compute the vector \(\overrightarrow{AB}\), subtract the coordinates of point \(A\) from those of \(B\):

• \(x\) component: \(3-1 = 2\)
• \(y\) component: \(2-1 = 1\)
• \(z\) component: \(7-3 = 4\)

This gives us \(\overrightarrow{AB} = (2, 1, 4)\). This method is then repeated to find \(\overrightarrow{CD}\), \(\overrightarrow{BC}\), and \(\overrightarrow{DA}\).

Vector subtraction helps in understanding the positioning and relationship between points in space, which is vital for determining shapes like parallelograms.

Scalar Multiplication

Scalar multiplication involves multiplying a vector by a scalar (a single real number). It results in altering the magnitude of the vector while maintaining its direction. Scalar multiplication is critical when determining parallel vectors because parallel vectors are scalar multiples of one another.

In the example for quadrilateral \(ABCD\), the vector \(\overrightarrow{CD} = (-2, -1, -4)\) is a scalar multiple of \(\overrightarrow{AB} = (2, 1, 4)\), specifically a factor of \(-1\). This demonstrates that \(\overrightarrow{CD}\) is parallel to \(\overrightarrow{AB}\). Similarly, \(\overrightarrow{BC}\) and \(\overrightarrow{DA}\) are parallel due to scalar multiplication by \(-1\).

Mastering scalar multiplication helps in identifying parallel vectors, a key factor in confirming the properties of shapes like parallelograms.

Vector Comparison

Vector comparison is about evaluating vectors to check if certain conditions, like parallelism or equality, are met. In the context of a parallelogram, the critical condition is whether vectors representing opposite sides are parallel, which involves comparing their direction and magnitude.

To determine if two vectors are parallel, we check if one can be expressed as a scalar multiple of the other. As seen in quadrilateral \(ABCD\), \(\overrightarrow{CD}\) is \(-1\) times \(\overrightarrow{AB}\), showing their parallel nature. Similarly, \(\overrightarrow{BC}\) is \(-1\) times \(\overrightarrow{DA}\), so they are parallel too. Checking that both pairs of opposite sides are parallel confirms the figure is a parallelogram.

Vector comparison is a powerful tool in geometry, facilitating the understanding of spatial relationships and the properties of geometric figures.