8. จงหาเวกเตอร์ที่ตั้งฉากทั้ง $$ \overline{\mathrm{u}} $$ และ $$ \overline{\mathrm{v}} $$ 1) $$ \overline{\mathrm{u}}=-3 \overline{\mathrm{i}}+2 \overline{\mathrm{j}}-2 \overline{\mathrm{k}} $$ และ $$ \overline{\mathrm{v}}=\overline{\mathrm{i}}-4 \overline{\mathrm{j}}+3 \overline{\mathrm{k}} $$ $$ \qquad $$ $$ \qquad $$ $$ \qquad $$ $$ \qquad $$ $$ \qquad $$ $$ \qquad $$ $$ \qquad $$ 2) $$ \overline{\mathbf{u}}=\overline{\mathrm{i}}+\overline{\mathrm{j}}-\overline{\mathrm{k}} $$ และ $$ \overline{\mathbf{v}}=\overline{\mathrm{j}} $$ $$ \qquad $$ $$ \qquad $$ $$ \qquad $$ $$ \qquad $$ $$ \qquad $$ $$ \qquad $$

Question

UpStudy AI · Accepted Answer

Alright, I need to find vectors that are perpendicular to both vectors **u** and **v**. Let's tackle the first problem first.

**Problem 1:**
Given:
- **u** = -3i + 2j - 2k
- **v** = i - 4j + 3k

I need to find a vector **w** such that **w** is perpendicular to both **u** and **v**. To find such a vector, I can use the cross product of **u** and **v**. The cross product of two vectors results in a vector that is perpendicular to both of them.

So, let's compute **u** × **v**.

**u** = (-3, 2, -2)
**v** = (1, -4, 3)

The cross product **u** × **v** is calculated as follows:

|i j k|
|-3 2 -2|
|1 -4 3|

Calculating the determinant:

i-component: (2*3) - (-2*-4) = 6 - 8 = -2
j-component: -[( -3*3 ) - ( -2*1 )] = -[-9 - (-2)] = -[-7] = 7
k-component: [(-3*-4) - (2*1)] = 12 - 2 = 10

So, **u** × **v** = -2i + 7j + 10k

Therefore, the vector **w** that is perpendicular to both **u** and **v** is **w** = -2i + 7j + 10k.

**Problem 2:**
Given:
- **u** = i + j - k
- **v** = j

I need to find a vector **w** such that **w** is perpendicular to both **u** and **v**.

First, let's express **v** in terms of i, j, and k:
**v** = 0i + 1j + 0k

Now, compute **u** × **v**.

**u** = (1, 1, -1)
**v** = (0, 1, 0)

|i j k|
|1 1 -1|
|0 1 0|

Calculating the determinant:

i-component: (1*0) - (-1*1) = 0 + 1 = 1
j-component: -[(1*0) - (-1*0)] = -[0 - 0] = 0
k-component: [(1*1) - (1*0)] = 1 - 0 = 1

So, **u** × **v** = 1i + 0j + 1k

Therefore, the vector **w** that is perpendicular to both **u** and **v** is **w** = i + 0j + k, which simplifies to **w** = i + k.

1) วekเตอร์ที่ตั้งฉากทั้ง **u** และ **v** คือ **w** = -2i + 7j + 10k 2) วekเตอร์ที่ตั้งฉากทั้ง **u** และ **v** คือ **w** = i + k

Upstudy AI Solution

Answer

Solution

The Deep Dive

Related Questions

Latest Geometry Questions