Problem 70
Given vec a\0,2λ,λ\ , vec b\2,2,1\ , vec c\-1,-2,-1\ and vec…
Given $\vec a\{0,2\lambda,\lambda\}$, $\vec b\{2,2,1\}$, $\vec c\{-1,-2,-1\}$ and $\vec d\{\lambda,0,1\}$. Determine $\lambda$ from the condition $\vec a\cdot\vec b=\vec c\cdot\vec d+7$.
Explanation of Method
Finding λ from a scalar-product condition.
Step 1: The two scalar products
\[ \vec a\cdot\vec b=0\cdot2+2\lambda\cdot2+\lambda\cdot1=5\lambda, \]
\[ \vec c\cdot\vec d=-\lambda+0-1=-\lambda-1. \]
Step 2: The condition
\[ 5\lambda=-\lambda-1+7\ \Longrightarrow\ 6\lambda=6. \]
\[\lambda=1\]
Step 3: Check
For $\lambda=1$: $\vec a=(0,2,1)$ and $\vec d=(1,0,1)$, so
\[ \vec a\cdot\vec b=0+4+1=5,\qquad \vec c\cdot\vec d+7=-2+7=5.\ \checkmark \]
| Quantity | Value |
|---|---|
| $\vec a\cdot\vec b$ | $5\lambda$ |
| $\vec c\cdot\vec d$ | $-\lambda-1$ |
| Equation | $5\lambda=6-\lambda$ |
| Solution | $\lambda=1$ |
$(1,5)$
$y=\vec a\cdot\vec b=5\lambda$
$y=\vec c\cdot\vec d+7$
- $\vec a\cdot\vec b=5\lambda$
- $\vec c\cdot\vec d+7=6-\lambda$
Result
\[\boxed{\,\lambda=1\,}\]