Math Problem Statement

For given vectors \( \vec{a} = 2\hat{i} - \hat{j} + 2\hat{k} \) and \( \vec{b} = -\hat{i} + \hat{j} - \hat{k} \), find the unit vector in the direction of the vector \( \vec{a} + \vec{b} \).

Solution

To solve this problem, we need to follow these steps:

Step 1: Define the Given Vectors

The given vectors are: a⃗=2i^−j^+2k^\vec{a} = 2\hat{i} - \hat{j} + 2\hat{k} b⃗=−i^+j^−k^\vec{b} = -\hat{i} + \hat{j} - \hat{k}

Step 2: Find a⃗+b⃗\vec{a} + \vec{b}

Add the components of a⃗\vec{a} and b⃗\vec{b} to find the resultant vector a⃗+b⃗\vec{a} + \vec{b}: a⃗+b⃗=(2−1)i^+(−1+1)j^+(2−1)k^\vec{a} + \vec{b} = (2 - 1)\hat{i} + (-1 + 1)\hat{j} + (2 - 1)\hat{k} Simplifying each component: a⃗+b⃗=i^+0j^+k^\vec{a} + \vec{b} = \hat{i} + 0\hat{j} + \hat{k} So, a⃗+b⃗=i^+k^\vec{a} + \vec{b} = \hat{i} + \hat{k}

Step 3: Find the Magnitude of a⃗+b⃗\vec{a} + \vec{b}

The magnitude of a⃗+b⃗\vec{a} + \vec{b} is calculated as follows: ∣a⃗+b⃗∣=(1)2+(0)2+(1)2|\vec{a} + \vec{b}| = \sqrt{(1)^2 + (0)^2 + (1)^2} ∣a⃗+b⃗∣=1+0+1=2|\vec{a} + \vec{b}| = \sqrt{1 + 0 + 1} = \sqrt{2}

Step 4: Find the Unit Vector in the Direction of a⃗+b⃗\vec{a} + \vec{b}

To find the unit vector in the direction of a⃗+b⃗\vec{a} + \vec{b}, divide each component of a⃗+b⃗\vec{a} + \vec{b} by its magnitude: Unit vector=a⃗+b⃗∣a⃗+b⃗∣=i^+k^2\text{Unit vector} = \frac{\vec{a} + \vec{b}}{|\vec{a} + \vec{b}|} = \frac{\hat{i} + \hat{k}}{\sqrt{2}} Thus, the unit vector is: 12i^+12k^=i^+k^2\frac{1}{\sqrt{2}}\hat{i} + \frac{1}{\sqrt{2}}\hat{k} = \frac{\hat{i} + \hat{k}}{\sqrt{2}}

Final Answer

The unit vector in the direction of a⃗+b⃗\vec{a} + \vec{b} is: i^+k^2\frac{\hat{i} + \hat{k}}{\sqrt{2}}

Would you like further clarification on any steps, or do you have more questions?

Additional Questions

  1. How would we calculate the unit vector if the magnitudes had complex numbers?
  2. Can you explain why we use the magnitude to find the unit vector?
  3. What if a⃗\vec{a} and b⃗\vec{b} had more dimensions, like in 4D space?
  4. How can we use this method for unit vectors in physics applications?
  5. Is there an alternative approach to find the unit vector?

Tip

Always ensure the magnitude is positive when finding unit vectors, as it represents the scalar length of a vector.

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Math Problem Analysis

Mathematical Concepts

Vector Addition
Unit Vector
Magnitude of a Vector

Formulas

Vector addition formula: \( \vec{a} + \vec{b} = (a_x + b_x)\hat{i} + (a_y + b_y)\hat{j} + (a_z + b_z)\hat{k} \)
Magnitude of vector \( \vec{v} = a\hat{i} + b\hat{j} + c\hat{k} \): \( |\vec{v}| = \sqrt{a^2 + b^2 + c^2} \)
Unit vector formula: \( \frac{\vec{v}}{|\vec{v}|} \)

Theorems

Properties of Vector Addition

Suitable Grade Level

Grades 10-12

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