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Given vectors: r = i + 2j − 4k, s = 4i − j + 3k, t = 2i + 3k. Calculate OP = r − 3s + t. What are the coordinates of P?

Mathematics

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Answers

2 parts
13
(-9, 5, -10)
14
8

Step-by-step solution

5 steps

Part 13· 3 steps

Given vectors:
r=i+2j−4k\mathbf{r} = \mathbf{i} + 2\mathbf{j} - 4\mathbf{k}
s=4i−j+3k\mathbf{s} = 4\mathbf{i} - \mathbf{j} + 3\mathbf{k}
t=2i+3k\mathbf{t} = 2\mathbf{i} + 3\mathbf{k}

  1. Calculate 3s3\mathbf{s}.

    3s=3(4i−j+3k)=12i−3j+9k3\mathbf{s} = 3(4\mathbf{i} - \mathbf{j} + 3\mathbf{k}) = 12\mathbf{i} - 3\mathbf{j} + 9\mathbf{k}

  2. Calculate OP⃗=r−3s+t\vec{OP} = \mathbf{r} - 3\mathbf{s} + \mathbf{t}.

    Substitute the vector values:

    OP⃗=(i+2j−4k)−(12i−3j+9k)+(2i+0j+3k)\vec{OP} = (\mathbf{i} + 2\mathbf{j} - 4\mathbf{k}) - (12\mathbf{i} - 3\mathbf{j} + 9\mathbf{k}) + (2\mathbf{i} + 0\mathbf{j} + 3\mathbf{k})

    Combine the i\mathbf{i} components:

    1−12+2=−91 - 12 + 2 = -9

    Combine the j\mathbf{j} components:

    2−(−3)+0=2+3=52 - (-3) + 0 = 2 + 3 = 5

    Combine the k\mathbf{k} components:

    −4−9+3=−13+3=−10-4 - 9 + 3 = -13 + 3 = -10

    So, the resultant vector is:

    OP⃗=−9i+5j−10k\vec{OP} = -9\mathbf{i} + 5\mathbf{j} - 10\mathbf{k}

  3. Determine the coordinates of P.

    Since O is the origin, the coordinates of P are the components of OP⃗\vec{OP}.
    The coordinates of P are (−9,5,−10)\boxed{(-9, 5, -10)}.

Part 148

Initial state:
Total beads = 14
Number of red beads = xx
Number of blue beads = 14−x14 - x

After removing 4 red beads:
Number of red beads = x−4x - 4
Total number of beads = 14−4=1014 - 4 = 10

The probability of selecting a red bead is given as 25\frac{2}{5}.

  1. Set up the probability equation.

    Number of red beadsTotal number of beads=25\frac{\text{Number of red beads}}{\text{Total number of beads}} = \frac{2}{5} x−410=25\frac{x - 4}{10} = \frac{2}{5}

  2. Solve for xx.

    Multiply both sides by 10:

    x−4=25×10x - 4 = \frac{2}{5} \times 10 x−4=4x - 4 = 4

    Add 4 to both sides:

    x=4+4x = 4 + 4 x=8x = 8

    The value of xx is 8\boxed{8}.

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