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Notely Maths 12
Class 12 Maths
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7.2 Q-2

Question Statement

Given vectors:

  • U‾=i‾−2j‾−k‾\underline{U} = \underline{\mathbf{i}} - 2 \underline{\mathbf{j}} - \underline{\mathbf{k}}
  • V‾=3i‾−2j+2k‾\underline{V} = 3 \underline{\mathbf{i}} - 2 \mathbf{j} + 2 \underline{k}
  • W‾=5i‾−j‾+3k‾\underline{W} = 5 \underline{\mathbf{i}} - \underline{\mathbf{j}} + 3 \underline{\mathbf{k}}

Find the following:

  1. (i) U‾+2V‾+W‾\underline{U} + 2 \underline{V} + \underline{W}
  2. (ii) V‾−3W‾\underline{V} - 3 \underline{W}
  3. (iii) ∣3V‾+W‾∣| 3 \underline{V} + \underline{W} |

Background and Explanation

In this problem, we are asked to perform operations on vectors. The key vector operations involved are:

  • Vector Addition: Add the corresponding components of two vectors.
  • Scalar Multiplication: Multiply each component of the vector by a scalar.
  • Magnitude of a Vector: The magnitude of a vector v=vxi+vyj+vzk\mathbf{v} = v_x \mathbf{i} + v_y \mathbf{j} + v_z \mathbf{k} is given by: ∣v∣=vx2+vy2+vz2|\mathbf{v}| = \sqrt{v_x^2 + v_y^2 + v_z^2}

These operations are fundamental in vector algebra and are necessary for solving the given problems.

Solution

(i) Find U‾+2V‾+W‾\underline{U} + 2 \underline{V} + \underline{W}

We begin by substituting the values of U‾\underline{U}, V‾\underline{V}, and W‾\underline{W} into the expression:

U‾+2V‾+W‾=(i‾−2j‾−k‾)+2(3i‾−2j‾+2k‾)+(5i‾−j‾+3k‾)\underline{U} + 2 \underline{V} + \underline{W} = (\underline{\mathbf{i}} - 2 \underline{\mathbf{j}} - \underline{\mathbf{k}}) + 2(3 \underline{\mathbf{i}} - 2 \underline{\mathbf{j}} + 2 \underline{\mathbf{k}}) + (5 \underline{\mathbf{i}} - \underline{\mathbf{j}} + 3 \underline{\mathbf{k}})

Now, let’s simplify each term:

  • 2V‾=2(3i‾−2j‾+2k‾)=6i‾−4j‾+4k‾2 \underline{V} = 2(3 \underline{\mathbf{i}} - 2 \underline{\mathbf{j}} + 2 \underline{\mathbf{k}}) = 6 \underline{\mathbf{i}} - 4 \underline{\mathbf{j}} + 4 \underline{\mathbf{k}}

Substitute back into the equation:

U‾+2V‾+W‾=i‾−2j‾−k‾+6i‾−4j‾+4k‾+5i‾−j‾+3k‾\underline{U} + 2 \underline{V} + \underline{W} = \underline{\mathbf{i}} - 2 \underline{\mathbf{j}} - \underline{\mathbf{k}} + 6 \underline{\mathbf{i}} - 4 \underline{\mathbf{j}} + 4 \underline{\mathbf{k}} + 5 \underline{\mathbf{i}} - \underline{\mathbf{j}} + 3 \underline{\mathbf{k}}

Now combine like terms:

  • For i\mathbf{i}: 1+6+5=12i1 + 6 + 5 = 12 \mathbf{i}
  • For j\mathbf{j}: −2−4−1=−7j-2 - 4 - 1 = -7 \mathbf{j}
  • For k\mathbf{k}: −1+4+3=6k-1 + 4 + 3 = 6 \mathbf{k}

Thus, the result is:

U‾+2V‾+W‾=12i‾−7j‾+6k‾\underline{U} + 2 \underline{V} + \underline{W} = 12 \underline{\mathbf{i}} - 7 \underline{\mathbf{j}} + 6 \underline{\mathbf{k}}

(ii) Find V‾−3W‾\underline{V} - 3 \underline{W}

Next, we subtract 3W‾3 \underline{W} from V‾\underline{V}:

3W‾=3(5i‾−j‾+3k‾)=15i‾−3j‾+9k‾3 \underline{W} = 3(5 \underline{\mathbf{i}} - \underline{\mathbf{j}} + 3 \underline{\mathbf{k}}) = 15 \underline{\mathbf{i}} - 3 \underline{\mathbf{j}} + 9 \underline{\mathbf{k}}

Now subtract:

V‾−3W‾=(3i‾−2j‾+2k‾)−(15i‾−3j‾+9k‾)\underline{V} - 3 \underline{W} = (3 \underline{\mathbf{i}} - 2 \underline{\mathbf{j}} + 2 \underline{\mathbf{k}}) - (15 \underline{\mathbf{i}} - 3 \underline{\mathbf{j}} + 9 \underline{\mathbf{k}})

Simplify by subtracting the components:

  • For i\mathbf{i}: 3−15=−12i3 - 15 = -12 \mathbf{i}
  • For j\mathbf{j}: −2−(−3)=−2+3=1j-2 - (-3) = -2 + 3 = 1 \mathbf{j}
  • For k\mathbf{k}: 2−9=−7k2 - 9 = -7 \mathbf{k}

Thus, the result is:

V‾−3W‾=−12i‾+j‾−7k‾\underline{V} - 3 \underline{W} = -12 \underline{\mathbf{i}} + \underline{\mathbf{j}} - 7 \underline{\mathbf{k}}

(iii) Find ∣3V‾+W‾∣| 3 \underline{V} + \underline{W} |

Now we calculate the magnitude of 3V‾+W‾3 \underline{V} + \underline{W}.

First, find 3V‾3 \underline{V}:

3V‾=3(3i‾−2j‾+2k‾)=9i‾−6j‾+6k‾3 \underline{V} = 3(3 \underline{\mathbf{i}} - 2 \underline{\mathbf{j}} + 2 \underline{\mathbf{k}}) = 9 \underline{\mathbf{i}} - 6 \underline{\mathbf{j}} + 6 \underline{\mathbf{k}}

Now add W‾\underline{W}:

3V‾+W‾=(9i‾−6j‾+6k‾)+(5i‾−j‾+3k‾)3 \underline{V} + \underline{W} = (9 \underline{\mathbf{i}} - 6 \underline{\mathbf{j}} + 6 \underline{\mathbf{k}}) + (5 \underline{\mathbf{i}} - \underline{\mathbf{j}} + 3 \underline{\mathbf{k}})

Combine like terms:

  • For i\mathbf{i}: 9+5=14i9 + 5 = 14 \mathbf{i}
  • For j\mathbf{j}: −6−1=−7j-6 - 1 = -7 \mathbf{j}
  • For k\mathbf{k}: 6+3=9k6 + 3 = 9 \mathbf{k}

Thus:

3V‾+W‾=14i‾−7j‾+9k‾3 \underline{V} + \underline{W} = 14 \underline{\mathbf{i}} - 7 \underline{\mathbf{j}} + 9 \underline{\mathbf{k}}

Finally, calculate the magnitude:

∣3V‾+W‾∣=(14)2+(−7)2+(9)2=196+49+81=326|3 \underline{V} + \underline{W}| = \sqrt{(14)^2 + (-7)^2 + (9)^2} = \sqrt{196 + 49 + 81} = \sqrt{326}

Key Formulas or Methods Used

  • Vector Addition: u+v=(ux+vx)i+(uy+vy)j+(uz+vz)k\mathbf{u} + \mathbf{v} = (u_x + v_x) \mathbf{i} + (u_y + v_y) \mathbf{j} + (u_z + v_z) \mathbf{k}

  • Scalar Multiplication: kv=(k⋅vx)i+(k⋅vy)j+(k⋅vz)kk \mathbf{v} = (k \cdot v_x) \mathbf{i} + (k \cdot v_y) \mathbf{j} + (k \cdot v_z) \mathbf{k}

  • Magnitude of a Vector: ∣v∣=vx2+vy2+vz2|\mathbf{v}| = \sqrt{v_x^2 + v_y^2 + v_z^2}

Summary of Steps

  1. (i) Add U‾\underline{U}, 2V‾2 \underline{V}, and W‾\underline{W} by performing component-wise addition.
  2. (ii) Subtract 3W‾3 \underline{W} from V‾\underline{V} by performing component-wise subtraction.
  3. (iii) Find the vector 3V‾+W‾3 \underline{V} + \underline{W}, then calculate its magnitude by applying the formula for the magnitude of a vector.