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Given vectors r1 = 2i + 2j + k and r2 = 6i − 3j + 2k, find the radius and the magnitude.

Mathematics

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Answer

3
7

42\sqrt{42}

Step-by-step solution

3 steps

Given the vectors:

r1=2i+2j+kr_1 = 2\mathbf{i} + 2\mathbf{j} + \mathbf{k} r2=6i−3j+2kr_2 = 6\mathbf{i} - 3\mathbf{j} + 2\mathbf{k}

The question asks for "the Radius and the magnitude". In the context of vectors, "magnitude" refers to the length of a vector. "Radius" is not a standard term for operations on two arbitrary vectors. We will interpret "the magnitude" as the magnitudes of the given vectors r1r_1 and r2r_2. For "the Radius", we will interpret it as the distance between the two points represented by the position vectors r1r_1 and r2r_2, which is the magnitude of the difference vector ∣r2−r1∣|r_2 - r_1|.

  1. Find the magnitude of r1r_1.

    The magnitude of a vector v=xi+yj+zk\mathbf{v} = x\mathbf{i} + y\mathbf{j} + z\mathbf{k} is given by ∣v∣=x2+y2+z2|\mathbf{v}| = \sqrt{x^2 + y^2 + z^2}.
    For r1=2i+2j+kr_1 = 2\mathbf{i} + 2\mathbf{j} + \mathbf{k}:

    ∣r1∣=22+22+12|r_1| = \sqrt{2^2 + 2^2 + 1^2} ∣r1∣=4+4+1|r_1| = \sqrt{4 + 4 + 1} ∣r1∣=9|r_1| = \sqrt{9} ∣r1∣=3|r_1| = 3

    The magnitude of r1r_1 is 3\boxed{3}.

  2. Find the magnitude of r2r_2.

    For r2=6i−3j+2kr_2 = 6\mathbf{i} - 3\mathbf{j} + 2\mathbf{k}:

    ∣r2∣=62+(−3)2+22|r_2| = \sqrt{6^2 + (-3)^2 + 2^2} ∣r2∣=36+9+4|r_2| = \sqrt{36 + 9 + 4} ∣r2∣=49|r_2| = \sqrt{49} ∣r2∣=7|r_2| = 7

    The magnitude of r2r_2 is 7\boxed{7}.

  3. Find the "Radius" (interpreted as the distance between the points represented by r1r_1 and r2r_2).

    First, calculate the difference vector r2−r1r_2 - r_1:

    r2−r1=(6i−3j+2k)−(2i+2j+k)r_2 - r_1 = (6\mathbf{i} - 3\mathbf{j} + 2\mathbf{k}) - (2\mathbf{i} + 2\mathbf{j} + \mathbf{k}) r2−r1=(6−2)i+(−3−2)j+(2−1)kr_2 - r_1 = (6-2)\mathbf{i} + (-3-2)\mathbf{j} + (2-1)\mathbf{k} r2−r1=4i−5j+kr_2 - r_1 = 4\mathbf{i} - 5\mathbf{j} + \mathbf{k}

    Now, find the magnitude of this difference vector:

    ∣r2−r1∣=42+(−5)2+12|r_2 - r_1| = \sqrt{4^2 + (-5)^2 + 1^2} ∣r2−r1∣=16+25+1|r_2 - r_1| = \sqrt{16 + 25 + 1} ∣r2−r1∣=42|r_2 - r_1| = \sqrt{42}

    The "Radius" (distance between the points) is 42\boxed{\sqrt{42}}.

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