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By the mid-afternoon drinks break, left-arm spinner Vettori had England reeling at 208 for six with two wickets in as many balls.
Paul Collingwood edged to slip and wicketkeeper Tim Ambrose shouldered arms to become the third lbw victim of the innings.
Although Stuart Broad survived the hat-trick delivery with a play and miss, England were in danger of a first-innings deficit.
But left-handers Broad and Ryan Sidebottom hung around long enough for Vaughan to more than double his score.
Rain and bad light sliced 153.2 overs off the opening three days' play and increased the probability of the match finishing in a draw - New Zealand were dismissed for 277 after being inserted on day one.
England v New Zealand
New Zealand 1st innings
JM How c Ambrose b Anderson 7
AJ Redmond c Cook b Anderson 0
JAH Marshall c Strauss b Broad 24
LRPL Taylor c Collingwood b Broad 19
BB McCullum b Panesar 97
DR Flynn b Anderson 9
JDP Oram c Strauss b Sidebottom28
DL Vettori b Sidebottom 48
KD Mills b Sidebottom 10
TG Southee b Sidebottom 1
CS Martin not out 0
Extras (b 16, lb 14, w 1, nb 3) 34
Total (all out; 86.2 overs) 277
Fall of wickets: 1-2 (Redmond), 2-18 (How), 3-41 (Taylor), 4-76 (Marshall), 5-104 (Flynn), 6-203 (McCullum), 7-222 (Oram), 8-258 (Mills), 9-260 (Southee), 10-277 (Vettori)
Bowling: RJ Sidebottom 28.2-12-55-4; JM Anderson 20-5-66-3; SCJ Broad 24-4-85-2; PD Collingwood 3-1-11-0; MS Panesar 11-2-30-1
England 1st innings
AJ Strauss lbw b Oram 63
AN Cook c McCullum b Martin 61
MP Vaughan c Marshall b Vettori 106
KP Pietersen lbw b Vettori 3
IR Bell c McCullum b Martin 16
PD Collingwood c Taylor b Vettori 6
TR Ambrose lbw b Vettori 0
SCJBroad b Oram 25
RJ Sidebottom c Taylor b Mills 16
MS Panesar c Flynn b Vettori 0
JM Anderson not out 0
Extras (b 3, lb 7, w 1, nb 12) 23
Total (all out; 111.3 overs) 319
Fall of wickets: 1-121 (Cook), 2-148 (Strauss), 3-152 (Pietersen), 4-180 (Bell), 5-208 (Collingwood), 6-208 (Ambrose), 7-269 (Broad), 8-317 (Sidebottom), 9-318 (Panesar), 10-319 (Vaughan).
Bowling: CS Martin 32-8-76-2; KD Mills 22-3-60-1; TG Southee 16-2-59-0; JDP Oram 19-5-45-2; DL Vettori 22.3-4-69-5.
New Zealand 2nd innings
JM How not out 26
AJ Redmond not out 14
Extras 0
Total (0 wickets; 15 overs) 40
To bat: JAH Marshall, DR Flynn, LRPL Taylor, JDP Oram, BB McCullum, DL Vettori, CS Martin, TG Southee, KD Mills
Bowling: RJ Sidebottom 4-2-7-0; JM Anderson 2-0-10-0; SCJ Broad 4-0-19-0; MS Panesar 5-2-4-0.
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