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For the points P(-2, 1, 3) and Q(3, 5, -1), in terms of i, j and k, if required, find a) PQ, b) |PQ|, c) PQ

Mathematics

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Here's the solution for question 7.

7. For the points P(-2, 1, 3) and Q(3, 5, -1), in terms of i\mathbf{i}, j\mathbf{j} and k\mathbf{k}, if required, find:

First, express the position vectors of P and Q:
P⃗=−2i+1j+3k\vec{P} = -2\mathbf{i} + 1\mathbf{j} + 3\mathbf{k}
Q⃗=3i+5j−1k\vec{Q} = 3\mathbf{i} + 5\mathbf{j} - 1\mathbf{k}

Part a· 2 steps

PQ⃗\vec{PQ}

  1. To find the vector PQ⃗\vec{PQ}, subtract the position vector of P from the position vector of Q.

    PQ⃗=Q⃗−P⃗=(3i+5j−k)−(−2i+j+3k)\vec{PQ} = \vec{Q} - \vec{P} = (3\mathbf{i} + 5\mathbf{j} - \mathbf{k}) - (-2\mathbf{i} + \mathbf{j} + 3\mathbf{k})

  2. Distribute the negative sign and combine the i\mathbf{i}, j\mathbf{j}, and k\mathbf{k} components.

    PQ⃗=(3−(−2))i+(5−1)j+(−1−3)k\vec{PQ} = (3 - (-2))\mathbf{i} + (5 - 1)\mathbf{j} + (-1 - 3)\mathbf{k} PQ⃗=(3+2)i+(4)j+(−4)k\vec{PQ} = (3 + 2)\mathbf{i} + (4)\mathbf{j} + (-4)\mathbf{k} PQ⃗=5i+4j−4k\vec{PQ} = 5\mathbf{i} + 4\mathbf{j} - 4\mathbf{k}

    The vector PQ⃗\vec{PQ} is \boxed{\text{5\mathbf{i} + 4\mathbf{j} - 4\mathbf{k}}}.

Part b· 2 steps

∣PQ⃗∣∗|\vec{PQ}|*

  1. To find the magnitude of PQ⃗\vec{PQ}, use the formula ∣PQ⃗∣=x2+y2+z2|\vec{PQ}| = \sqrt{x^2 + y^2 + z^2}, where PQ⃗=xi+yj+zk\vec{PQ} = x\mathbf{i} + y\mathbf{j} + z\mathbf{k}.

    From part (a), x=5x=5, y=4y=4, and z=−4z=-4.

    ∣PQ⃗∣=(5)2+(4)2+(−4)2|\vec{PQ}| = \sqrt{(5)^2 + (4)^2 + (-4)^2}

  2. Calculate the squares and sum them.

    ∣PQ⃗∣=25+16+16|\vec{PQ}| = \sqrt{25 + 16 + 16} ∣PQ⃗∣=57|\vec{PQ}| = \sqrt{57}

    The magnitude ∣PQ⃗∣|\vec{PQ}| is \boxed{\text{\sqrt{57}}}.

Part c· 2 steps

PQ^\widehat{PQ}

  1. To find the unit vector PQ^\widehat{PQ} in the direction of PQ⃗\vec{PQ}, divide the vector PQ⃗\vec{PQ} by its magnitude ∣PQ⃗∣|\vec{PQ}|.

    PQ^=PQ⃗∣PQ⃗∣=5i+4j−4k57\widehat{PQ} = \frac{\vec{PQ}}{|\vec{PQ}|} = \frac{5\mathbf{i} + 4\mathbf{j} - 4\mathbf{k}}{\sqrt{57}}

  2. Express the unit vector with each component divided by the magnitude.

    PQ^=557i+457j−457k\widehat{PQ} = \frac{5}{\sqrt{57}}\mathbf{i} + \frac{4}{\sqrt{57}}\mathbf{j} - \frac{4}{\sqrt{57}}\mathbf{k}

    The unit vector PQ^\widehat{PQ} is \boxed{\text{\frac{5}{\sqrt{57}}\mathbf{i} + \frac{4}{\sqrt{57}}\mathbf{j} - \frac{4}{\sqrt{57}}\mathbf{k}}}.

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